# Introduction

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

In this article, we will 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 concerning, for example, such an important concept as the center of mass of a 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. The integration of the equations of motion of this children's toy is one of the still unsolved problems of mechanics ...

# 1. Old as the world, the d'Alembert principle To begin, consider the movement of the material point. Directly from the axioms, the basic equation for the dynamics of a point is accelerated times mass is the vector sum of the forces applied to the point. And the forces that are applied to the point need to talk in more detail. In a section of mechanics called analytical mechanics, the forces applied to points in a mechanical system are subject to strict classification.

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

1. Active forces . This group of forces can be given the following definition
Active forces are called, the value of which can be determined from the conditions

In formal language, the active force is determined by the vector function where is the generalized coordinate of the point; Is the generalized velocity of a point. From this expression it is clear that starting to solve the problem of motion and having the initial conditions (time, position and speed), you can immediately calculate the active force.

The force of gravity, elasticity, the Coulomb force of interaction of a charge with an electric field, the Ampere force and the Lorentz force, the force of viscous friction and aerodynamic drag are all examples of active forces. The expressions for their calculation are known and these forces can be calculated by knowing the position and speed of the point.
2. Bond reactions . The most unpleasant forces that you can think of. Let me remind you of one of the axioms of statics, called the axiom of relations
The bonds applied to the body can be discarded by replacing their action with force, or a system of forces

The dot shown in the figure is not a free dot. Its movement is limited by a bond, conventionally represented as a certain surface, within which the trajectory of movement is located. The above axiom makes it possible to remove the surface by applying a force to the point whose action is equivalent to the presence of the surface. Moreover, this force is not known in advance - its value satisfies the constraints on position, speed and acceleration imposed by the connection, and of course, the reaction vector depends on the applied active forces. Relationship reactions are to be determined in the process of solving the problem. Dry friction also relates to bond reactions, the presence of which even in a simple task significantly complicates the process of solving it.

Based on this classification, the equation of motion of the point (1) is rewritten in the form where is the resultant of the active forces applied to the point; - the resultant of the reactions superimposed on the connection point.

And now we’ll do the simplest trick - we transfer the acceleration with mass to another part of equation (2) and introduce the notation Then, equation (2) turns into The force represented by vector (3) is called the d'Alembert force of inertia . And equation (4) expresses the d'Alembert principle for a material point

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

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

A fairly widespread debate is whether inertia 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 persistent) acceleration, which bends the trajectory of a point. My personal opinion is that the forces of inertia are the mathematical focus shown above, which allows us to move on to considering equilibrium instead of accelerated motion. The inertia force (3) is determined by the acceleration of the point, but it, in turn, is determined by the action of the forces applied to the point, and in accordance with Newton’s axiomatics, the force is primary. Therefore, we cannot speak of any “physicality” of inertia forces. Nature does not know the active forces that depend on acceleration.

# 2. The d'Alembert principle for a solid. The main vector and the main moment of inertia forces Now we extend equation (4) to the case of the motion of a rigid body. In mechanics, it is considered as an immutable mechanical system consisting of many 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 particular point can be divided into external actives , reactions of external relations , and internal forces , which are the forces of interaction of the considered point with other points of the body (in fact - internal reactions). All the forces mentioned are the resultants of the corresponding group of forces applied to the point. We apply to this equation the d'Alembert principle where - the force of inertia applied to a given point in the body.

Now that all the points of the body are in equilibrium, we can use the equilibrium condition of the solid, which gives us 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 selected center O, are wound to zero The main vector of the system of forces is the vector sum of all the 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 up the equations for all points, in its left part we get the main vector Moreover, the sum of 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 the principal moment of the system of forces equal to zero 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 as applied to a solid, but with only one necessary correction.

The number of active forces and bond reactions in equations (5) and (6) is finite. Most of the terms in the corresponding sums are equal to zero, because active external forces and reactions of external relations, generally speaking, are applied only at some 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 inertia forces, and the sum of their moments relative to the selected center is the sum of the integral. The system of inertia forces is usually reduced to the main vector and the main moment, and we can write that the main vector and the main moment of the inertia forces applied to the solid. The integrals (7) and (8) are taken over the entire volume of the body, and is the radius of the 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 d'Alembert principle for a solid
The solid body is in equilibrium under the action of external forces applied to it, bond reactions, the main vector and the main moment of inertia forces.

In essence, (10) and (11) is a form of writing 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 equations of motion is not the most convenient. After all, integrals (7) and (8) can be calculated in a general form and come to more convenient equations of motion. In this regard, (10) and (11) should be considered as the theoretical basis for constructing analytical mechanics.

# 3. The center of mass and the inertia tensor enter the stage.

Let us return to our tensors and use them to calculate the integrals (7) and (8) for the general case of the motion of a rigid body. As the center of reduction, we choose the point O 1 . This point is selected as a pole and the local basis of the coordinate system associated with the body is defined in it. In one of the previous articles, we determined the tensor relation for the acceleration of a body point in such a motion Multiplying (12) by the mass of the point with a minus sign, we obtain the inertia force applied to the solid volume element 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 the pseudovectors of angular velocity and angular acceleration We substitute (14) into (13) and take the triple integral over the entire volume of the body, taking into account that the angular velocity and angular acceleration are the same at each point of this volume, that is, they can be taken outside the sign of the integral The integral in the first term is body weight. The integral in the second term is a more interesting thing. Recall one of the formulas of the course of theoretical mechanics: where are the contravariant components of the radius vector of the center of mass of the body in question. Without going into the meaning of the concept of the center of mass, we simply replace the integrals in (15) in accordance with formula (16), taking into account that covariant components are used in the second term (15). Yeah, expression (17) is also familiar to us, we will present it in a more familiar vector form The first term in (18) is the inertia force associated with the translational motion of the body along with the pole. The second term is the centrifugal inertia force associated with the tangible 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 obtain this expression if it would be obtained in a vector form in an equally obvious way?” In response, I will say that getting formulas (17) and (18) was a warm-up. Now we get the expression of the main moment of inertia forces with respect to the selected pole, and here the tensor approach manifests itself in all its glory.

Take equation (13) and multiply it vectorwise from the left by the radius of the vector of the point of the body relative to the pole. Thus, we get the moment of inertia applied to the elementary volume of the body Again we perform the substitution (14) in (19), but do not rush to take the integral I don’t know about you, but it’s ruffling in my eyes, even when I am used to such formulas. The terms are arranged in a more natural order - the rotational and centrifugal components are rearranged. In addition, the complexity of transformative calculations increases from the first term to the second. We will simplify them one at a time, first we simplify the first, immediately taking the integral Then the radius of the center of mass vector appeared again. There is nothing complicated here - we have one acceleration of the pole and we carried it out of the integral sign. We will deal with the interpretation a little later, but for now we will transform the second term (20). In it, we can perform a convolution of the product of Levi-Civita tensors by the mute index k Here we took advantage of the Kronecker delta property 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 whole body volume What time! The obscure "crocodile", by means of formal tensor transformations, collapsed into a compact formula. 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 the tensor of inertia of a solid . This value is truly fundamental for mechanics, and we will talk about it in more detail, for now I will only say that (24) is a tensor of the second rank, whose components are the axial and centrifugal moments of inertia of the body in the selected coordinate system. It characterizes the inertia of a solid during rotation. I draw the reader’s attention to how quickly we got an expression for the inertia tensor, essentially acting in a formal way. With vector ratios you can’t do without breaking the brain, I was convinced of this from personal experience.

Finally, we turn to the last term (20). When taking the integral in it, the inertia tensor should also be obtained, 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 by folding 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 it is a vector product . But the inertia tensor in (25) is not visible. In order to obtain it, we carry out a series of equivalent transformations Here we again took into account that we 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 forces associated with centrifugal forces Expression (27) is equivalent to the vector-matrix relation: And although pathos phrases overwhelm me, 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, the main vector and the main moment of the inertia forces applied to the solid are And now we’ll admire it — despite the fact that the above transformations are similar to Egyptian hieroglyphs, they are formal , we just performed actions on tensor indices and used the properties of tensor operations. We did not need 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 the final expression came out automatically. In addition, such fundamental characteristics as the coordinates of the center of mass of the body and the inertia tensor are naturally obtained.

Reading lectures to students, I set out to derive (29) and (30) in terms of vectors. After I translated the stack of paper and pretty much breaking my brains, I came to the result. Take a word - the above transformations are just seeds, in comparison with 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, for which we took the pole O 1 . These expressions will help us understand what the center of mass of the body is 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, after substituting, we arrive at the differential equations of motion of a rigid body Why are these equations bad? 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, angular acceleration - on the acceleration of the pole. The vector determines the position of the center of mass of the body relative to the pole. But what if we choose the pole right in the center of mass? Then, after all , 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 Euler dynamic equation of spherical motion. And these equations are independent of each other. Thus, the center of mass of a solid is the point relative to which the inertia forces are reduced to the simplest form. The forward movement along with the pole and the spherical around the pole are dynamically decoupled. The inertia tensor of the body, calculated relative to the center of mass, is called the central inertia tensor .

Equations (33), (34) in the foreign literature are called the Newton-Euler equations, and are currently very actively used to build software designed for modeling mechanical systems. As part of the tensor cycle, we will recall them more than once.

# Conclusion

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

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

To be continued...