All methods for solving problems of dynamics that we have considered so far are based on equations that follow either directly from Newton's laws, or from general theorems that are consequences of these laws. However, this path is not the only one. It turns out that the equations of motion or the equilibrium conditions of a mechanical system can be obtained by assuming other general propositions instead of Newton's laws, called the principles of mechanics. In a number of cases, the application of these principles makes it possible, as we shall see, to find more efficient methods for solving the corresponding problems. In this chapter, one of the general principles of mechanics, called d'Alembert's principle, will be considered.
Suppose we have a system consisting of n material points. Let's single out some of the points of the system with mass . Under the action of external and internal forces applied to it and (which include both active forces and coupling reactions), the point receives some acceleration with respect to the inertial reference frame.
Let us introduce into consideration the quantity
having the dimension of force. A vector quantity equal in absolute value to the product of the mass of a point and its acceleration and directed opposite to this acceleration is called the force of inertia of the point (sometimes the d'Alembert force of inertia).
Then it turns out that the movement of a point has the following general property: if at each moment of time we add the force of inertia to the forces actually acting on the point, then the resulting system of forces will be balanced, i.e. will
.
This expression expresses the d'Alembert principle for one material point. It is easy to see that it is equivalent to Newton's second law and vice versa. Indeed, Newton's second law for the point in question gives 
. Transferring the term here to the right side of the equality, we arrive at the last relation.
Repeating the above reasoning with respect to each of the points of the system, we arrive at the following result, which expresses the d'Alembert principle for the system: if at any moment of time to each of the points of the system, in addition to the external and internal forces actually acting on it, the corresponding inertia forces are applied, then the resulting system of forces will be in equilibrium and all the equations of statics can be applied to it.
The significance of the d'Alembert principle lies in the fact that when it is directly applied to problems of dynamics, the equations of motion of the system are compiled in the form of well-known equilibrium equations; which makes a uniform approach to solving problems and usually greatly simplifies the corresponding calculations. In addition, in conjunction with the principle of possible displacements, which will be discussed in the next chapter, the d'Alembert principle allows us to obtain a new general method for solving problems of dynamics.
Applying the d'Alembert principle, it should be borne in mind that only external and internal forces act on a point of a mechanical system, the movement of which is being studied, and , arising as a result of the interaction of the points of the system with each other and with bodies that are not included in the system; under the action of these forces, the points of the system and move with the corresponding accelerations. The forces of inertia, which are mentioned in the d'Alembert principle, do not act on moving points (otherwise, these points would be at rest or move without acceleration, and then there would be no inertial forces themselves). The introduction of inertial forces is just a technique that allows you to compose the equations of dynamics using simpler methods of statics.
It is known from statics that the geometric sum of forces in equilibrium and the sum of their moments with respect to any center ABOUT are equal to zero, and according to the principle of solidification, this is true for forces acting not only on a rigid body, but also on any variable system. Then, on the basis of the d'Alembert principle, it should be.
Initially, the idea of this principle was expressed by Jacob Bernoulli (1654-1705) when considering the problem of the center of oscillation of bodies of arbitrary shape. In 1716, the St. Petersburg academician Ya. German (1678 - 1733) put forward the principle of static equivalence of "free" movements and "actual" movements, that is, movements carried out in the presence of connections. Later, this principle was applied by L. Euler (1707-1783) to the problem of vibrations of flexible bodies (the work was published in 1740) and was called the "Petersburg principle". However, the first to formulate the principle under consideration in a general form, although he did not give it a proper analytical expression, was d'Alembert (1717-1783). In his "Dynamics" published in 1743, he indicated a general method of approach to solving the problems of the dynamics of non-free systems. An analytical expression of this principle was later given by Lagrange in his Analytical Mechanics.
Consider some non-free mechanical system. Let us denote the resultant of all active forces acting on any point of the system through and the resultant of the reactions of the bonds - through Then the equation of motion of the point will have the form
where is the acceleration vector of a point, and is the mass of this point.
If we introduce into consideration a force called the d'Alembert force of inertia, then the equation of motion (2.9) can be rewritten in the form of an equation for the equilibrium of three forces:
Equation (2.10) is the essence of the d'Alembert principle for a point, and the same equation, extended to a system, is the essence of the d'Alembert principle for a system.
The equation of motion, written in the form (2.10), allows us to give the d'Alembert principle the following formulation: if the system is in motion, at some point in time, instantly stop and apply to each material point of this system the active reaction forces acting on it at the moment of stopping and d'Alembert forces of inertia, then the system will remain in equilibrium.
The d'Alembert principle is a convenient methodical method for solving dynamic problems, since it allows the equations of motion of non-free systems to be written in the form of static equations.
By this, of course, the problem of dynamics is not reduced to the problem of statics, since the problem of integrating the equations of motion is still preserved, but the d'Alembert principle provides a unified method for compiling the equations of motion of non-free systems, and this is its main advantage.
If we keep in mind that reactions are the action of bonds on the points of the system, then the d'Alembert principle can also be given the following formulation: if we add the d'Alembert forces of inertia to the active forces acting on the points of a non-free system, then the resulting forces of these forces will be balanced by the reactions of the bonds. It should be emphasized that this formulation is arbitrary, since in reality
when the system moves, there is no balancing, since the forces of inertia are not applied to the points of the system.
Finally, the d'Alembert principle can be given one more equivalent formulation, for which we rewrite equation (2.9) in the following form:
The d'Alembert principle establishes a unified approach to the study of the movement of a material object, regardless of the nature of the conditions imposed on this movement. In this case, the dynamic equations of motion are given the form of equilibrium equations. Hence the second name of the d'Alembert principle is the method of kinetostatics.
For a material point at any moment of motion, the geometric sum of the applied active forces, the reactions of the bonds and the conditionally attached inertia force is zero (Fig. 48).
Where Ф is the force of inertia of a material point, equal to:
.
(15.2)
Figure 48
Figure 49
The force of inertia is applied not to a moving object, but to the bonds that determine its movement. Man reports acceleration 
trolley (Fig. 49), pushing it with force 
.The force of inertia is the counteraction to the action of a person on the trolley, i.e. modulo equal to force 
and directed in the opposite direction.
If a point moves along a curved path, then the force of inertia can be projected onto the natural coordinate axes.
Figure 50
;
(15.3)
, (15.4) where 
-- radius of curvature of the trajectory.
When solving problems using the kinetostatics method, it is necessary to:
1. choose a coordinate system;
2. show all active forces applied to each point;
3. discard connections, replacing them with appropriate reactions;
4. add the force of inertia to the active forces and reactions of the bonds;
5. compose the equations of kinetostatics, from which to determine the desired values.
EXAMPLE 21.
ABOUT
SOLUTION. 1. Consider a car at the top of a convex bridge. Consider the car as a material point on which the given force 2. Since the car is moving at a constant speed, we write down the d'Alembert principle for a material point in projection onto the normal 
and communication reaction 
.
. (1) We express the force of inertia: 
; we determine the normal pressure of the car from equation (1): N.
\u003d 20m and moving at a constant speed V \u003d 36 km / h (Fig. 51).
16. The d'Alembert principle for a mechanical system. Principal vector and principal moment of inertia forces.
If to each point of the mechanical system at any moment of motion the corresponding inertia forces are conditionally applied, then at any moment of motion the geometric sum of the active forces acting on the point, the reactions of the bonds and the force of inertia is equal to zero.
The equation expressing the d'Alembert principle for a mechanical system has the form 
. (16.1) The sum of the moments of these balanced forces relative to any center is also equal to zero 
. (16.2) When applying the d'Alembert principle, the equations of motion of the system are compiled in the form of equilibrium equations. Equations (16.1) and (16.2) can be used to determine dynamic responses.
EXAMPLE 22.
Vertical shaft AK, rotating at a constant angular velocity 
\u003d 10s -1, fixed with a thrust bearing at point A and a cylindrical bearing at point K (Fig. 52). A thin homogeneous broken rod with a mass m=10kg and a length 10b is attached to the shaft at point E, consisting of parts 1 and 2, where b=0.1m, and their masses m 1 and m 2 are proportional to the lengths. The rod is attached to the shaft by a hinge at point E and a weightless rod 4 rigidly fixed at point B. Determine the reaction of hinge E and rod 4.
SOLUTION.
1. The length of the broken rod is 10b. Let's express the masses of the parts of the rod, proportional to the lengths: m 1 =0.4m; m 2 =0.3m; m 3 \u003d 0.3m.
Figure 42
2. To determine the desired reactions, consider the motion of a broken rod and apply the d'Alembert principle. Let's place the rod in the xy plane, depict the external forces acting on it: 
,
,
, hinge reactions 
And 
and reaction 
rod 4. We add to these forces the inertia forces of the parts of the rod: 
;
;
,
Where 
;
;
.
Then N.N.N.
Line of action of the resultant forces of inertia 
,
And 
passes at distances h 1 , h 2 and h 3 from the x-axis: m;
3. According to the d'Alembert principle, the applied active forces, the reactions of the bonds and the forces of inertia form a balanced system of forces. Let us compose three equilibrium equations for a flat system of forces:
;
;
(1)
;;
(2)
;.(3)
Solving the system of equations (1) + (3), substituting the given values of the corresponding quantities, we find the desired reactions:
N= yE=xE=
If all the forces acting on the points of a mechanical system are divided into external 
and internal 
, (Fig. 53), then for an arbitrary point of the mechanical system, two vector equalities can be written:
;
(16.3)
.
Figure 53
Taking into account the properties of internal forces, we obtain the d'Alembert principle for a mechanical system in the following form: 
;
(16.4)
, (16.5) where 
,
-- respectively, the main vectors of external forces and inertia forces;
,
- respectively, the main moments of external forces and inertia forces relative to an arbitrary center O.
Main vector 
and main point 
replace the inertial forces of all points of the system, since it is necessary to apply its own inertia force to each point of the system, depending on the acceleration of the point. Using the theorem on the motion of the center of mass and on the change in the angular momentum of the system relative to an arbitrary center, we obtain: 
,
(16.6)
. (16.7) For a rigid body rotating around a fixed axis z, the main moment of inertia about this axis is equal to 
, (16.8) where 
is the angular acceleration of the body.
During the translational motion of the body, the inertial forces of all its points are reduced to the resultant, equal to the main vector of inertial forces, i.e. 
.
P
Figure 54
, (16.9) where 
- the moment of inertia of the body about the axis of rotation.
If the body has a plane of symmetry and rotates around a fixed axis z, perpendicular to the plane of symmetry and not passing through the center of mass of the body, the force of inertia of all points of the body is reduced to the resultant, equal to the main vector of the forces of inertia of the system, but applied to some point K (Fig. 54) . Line of action of the resultant 
away from point O at a distance 
.
(16.10)
With a plane motion of a body having a plane of symmetry, the body moves along this plane (Fig. 55). The main vector and the main moment of the forces of inertia also lie in this plane and are determined by the formulas:
Figure 55
;
.
The minus sign indicates that the direction of the moment 
opposite to the direction of the angular acceleration of the body.
EXAMPLE 23.
Determine the force that tends to break a uniformly rotating flywheel of mass m, considering its mass distributed over the rim. Flywheel radius r, angular velocity 
(Fig. 56).
SOLUTION.
1. Seeking strength 
is internal. 
-- the resultant of the forces of inertia of the elements of the rim. 
. We express the x coordinate from the center of mass of the rim arc with a central angle 
:
, Then 
.
2. To determine the strength 
apply the d'Alembert principle in projection onto the x-axis: 
;
, where 
.
3. If the flywheel is a solid homogeneous disk, then 
, Then 
.
The scope of d'Alembert's principle is the dynamics of non-free mechanical systems. d'Alembert proposed an original method for solving problems of dynamics, which makes it possible to use fairly simple equations of statics. He wrote: "This rule reduces all problems related to the motion of bodies to simpler problems of equilibrium."
This method is based on the forces of inertia. Let's introduce this concept.
The force of inertia is called the geometric sum of the forces of counteraction of a moving material particle to bodies that impart acceleration to it.
Let us explain this definition. On fig. 15.1 shows a material particle M , interacting with n material objects. On fig. 15.1 shows the forces of interaction: without
which are actually not per particle, but on bodies with masses m 1 , …, m n . It is clear that the resultant of this system of converging reaction forces, R'=ΣF'k , modulo equal to R and is directed opposite to the acceleration, i.e.: R' = -ma. This force is the force of inertia referred to in the definition. In what follows, we will denote it by the letter F , i.e.:
In the general case of curvilinear motion of a point, acceleration is the sum of two components:
From (15.4) it can be seen that the components of the inertia force are directed oppositely to the directions of the corresponding components of the acceleration of the point. The modules of the components of the inertial force are determined by the following formulas:
Where ρ is the radius of curvature of the point trajectory.
After determining the force of inertia, consider d'Alembert's principle.
Let a mechanical system consisting of n material points (Fig. 15.2). Let's take one of them. All forces acting on k -th point, we classify into groups:
Expression (15.6) reflects the essence of the d'Alembert principle, written for one material point. By repeating the above steps with respect to each point of the mechanical system, we can write the system n equations similar to (15.6), which will be the mathematical record of the d'Alembert principle as applied to a mechanical system. Thus, we formulate d'Alembert's principle for a mechanical system:
If at any moment of time, in addition to the external and internal forces actually acting on it, an appropriate inertia force is applied to each point of a mechanical system, then the entire system of forces will be brought into equilibrium and all the equations of statics can be applied to it.
Keep in mind:
The d'Alembert principle can be applied to dynamic processes occurring in
inertial reference systems. The same requirement, as noted earlier, should be adhered to when applying the laws of dynamics;
The forces of inertia, which, according to the methodology of the d'Alembert principle, must be applied
live to the points of the system, in fact they are not affected. Indeed, if they existed, then the entire set of forces applied to each point would be in equilibrium, and the formulation of the problem of dynamics itself would be absent.
For an equilibrium system of forces, the following equations can be written:
those. the geometric sum of all forces of the system, including the forces of inertia, and the geometric sum of the moments of all forces about an arbitrary center are equal to zero.
Given the properties of the internal forces of the system:
expressions (15.7) can be significantly simplified.
Introducing the principal vector notation
and main point
expressions (15.7) will appear in the form:
Equations (15.11) are a direct continuation of the d'Alembert principle, but do not contain internal forces, which is their undoubted advantage. Their use is most effective in studying the dynamics of mechanical systems consisting of rigid bodies.
If we consider a system that consists of several material points, highlighting one specific point with a known mass, then under the action of external and internal forces applied to it, it receives some acceleration relative to the inertial reference frame. Among such forces there can be both active forces and coupling reactions.
The force of inertia of a point is a vector quantity, which is equal in absolute value to the product of the mass of the point and its acceleration. This value is sometimes referred to as the d'Alembert force of inertia, it is directed opposite to the acceleration. In this case, the following property of a moving point is revealed: if at each moment of time we add the force of inertia to the forces actually acting on the point, then the resulting system of forces will be balanced. So it is possible to formulate d'Alembert's principle for one material point. This statement is fully consistent with Newton's second law.
d'Alembert's principles for the system
If we repeat all the arguments for each point in the system, they lead to the following conclusion, which expresses the d’Alembert principle formulated for the system: if at any time we apply to each of the points in the system, in addition to the actually acting external and internal forces, then this system will be in equilibrium, so all the equations that are used in statics can be applied to it.
If we apply the d'Alembert principle to solve problems of dynamics, then the equations of motion of the system can be compiled in the form of the equilibrium equations known to us. This principle greatly simplifies calculations and makes the approach to solving problems unified.
Application of the d'Alembert principle
It should be taken into account that only external and internal forces act on a moving point in a mechanical system, which arise as a result of the interaction of points between themselves, as well as with bodies that are not included in this system. Points move with certain accelerations under the influence of all these forces. The forces of inertia do not act on moving points, otherwise they would move without acceleration or be at rest.
The forces of inertia are introduced only in order to compose the equations of dynamics using simpler and more convenient methods of statics. It is also taken into account that the geometric sum of internal forces and the sum of their moments is equal to zero. The use of equations that follow from the d'Alembert principle makes the process of solving problems easier, since these equations no longer contain internal forces.
