Necessary conditions for equilibrium of a mechanical system. Statics

Mechanical balance

Mechanical balance- a state of a mechanical system in which the sum of all forces acting on each of its particles is equal to zero and the sum of the moments of all forces applied to the body relative to any arbitrary axis of rotation is also zero.

In a state of equilibrium, the body is at rest (the velocity vector is zero) in the chosen reference frame, either moves uniformly in a straight line or rotates without tangential acceleration.

Definition through system energy

Since energy and forces are related by fundamental relationships, this definition is equivalent to the first. However, the definition in terms of energy can be extended to provide information about the stability of the equilibrium position.

Types of balance

Let's give an example for a system with one degree of freedom. In this case, a sufficient condition for the equilibrium position will be the presence of a local extremum at the point under study. As is known, the condition for a local extremum of a differentiable function is that its first derivative is equal to zero. To determine when this point is a minimum or maximum, you need to analyze its second derivative. The stability of the equilibrium position is characterized by the following options:

• unstable equilibrium;
• stable balance;
• indifferent equilibrium.

Unstable equilibrium

In the case when the second derivative is negative, the potential energy of the system is in a state of local maximum. This means that the equilibrium position unstable. If the system is displaced a small distance, it will continue its movement due to the forces acting on the system.

Stable balance

Second derivative > 0: potential energy at local minimum, equilibrium position sustainable(see Lagrange's theorem on the stability of equilibrium). If the system is displaced a small distance, it will return back to its equilibrium state. Equilibrium is stable if the center of gravity of the body occupies the lowest position compared to all possible neighboring positions.

Indifferent Equilibrium

Second derivative = 0: in this region the energy does not vary and the equilibrium position is indifferent. If the system is moved a small distance, it will remain in the new position.

Stability in systems with a large number of degrees of freedom

If a system has several degrees of freedom, then it may turn out that in shifts in some directions the equilibrium is stable, but in others it is unstable. The simplest example of such a situation is a “saddle” or “pass” (it would be good to place a picture in this place).

The equilibrium of a system with several degrees of freedom will be stable only if it is stable in all directions.

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mechanical balance- mechaninė pusiausvyra statusas T sritis fizika atitikmenys: engl. mechanical equilibrium vok. mechanisches Gleichgewicht, n rus. mechanical equilibrium, n pranc. équilibre mécanique, m … Fizikos terminų žodynas

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Equilibrium of a mechanical system- this is a state in which all points of a mechanical system are at rest with respect to the reference system under consideration. If the reference frame is inertial, equilibrium is called absolute, if non-inertial - relative.

To find the equilibrium conditions absolutely solid it is necessary to mentally break it down into a large number of fairly small elements, each of which can be represented by a material point. All these elements interact with each other - these interaction forces are called internal. In addition, external forces can act on a number of points on the body.

According to Newton's second law, for the acceleration of a point to be zero (and the acceleration of a point at rest to be zero), the geometric sum of the forces acting on that point must be zero. If a body is at rest, then all its points (elements) are also at rest. Therefore, for any point of the body we can write:

where is the geometric sum of all external and internal forces acting on i th element of the body.

The equation means that for a body to be in equilibrium, it is necessary and sufficient that the geometric sum of all forces acting on any element of this body be equal to zero.

From this it is easy to obtain the first condition for the equilibrium of a body (system of bodies). To do this, it is enough to sum up the equation for all elements of the body:

.

The second sum is equal to zero according to Newton's third law: the vector sum of all internal forces of the system is equal to zero, since any internal force corresponds to a force equal in magnitude and opposite in direction.

Hence,

.

The first condition for the equilibrium of a rigid body(systems of bodies) is the equality to zero of the geometric sum of all external forces applied to the body.

This condition is necessary, but not sufficient. This is easy to verify by remembering the rotating action of a pair of forces, the geometric sum of which is also zero.

The second condition for the equilibrium of a rigid body is the equality to zero of the sum of the moments of all external forces acting on the body relative to any axis.

Thus, the equilibrium conditions of a rigid body in the case of an arbitrary number of external forces look like this:

.

I'll consider material point, the movement of which is limited in such a way that it has only one degree of freedom.

This means that its position can be determined using a single quantity, such as the x coordinate. An example is a ball sliding without friction along a fixed wire bent in a vertical plane (Fig. 26.1a).

Another example is a ball attached to the end of a spring, sliding without friction to a horizontal guide (Fig. 26.2, a).

A conservative force acts on the ball: in the first case it is the force of gravity, in the second case it is the elastic force of a deformed spring. Potential energy graphs are shown in Fig. 26.1, b and 26.2, b.

Since the balls move along the wire without friction, the force with which the wire acts on the ball is in both cases perpendicular to the speed of the ball and, therefore, does no work on the ball. Therefore, energy conservation takes place:

From (26.1) it follows that, kinetic energy can only increase due to a decrease in amusing energy. Therefore, if the ball is in such a state that its speed is zero and the potential energy has a minimum value, then without external influence it will not be able to move, i.e. it will be in equilibrium.

The minima of U correspond to equal values ​​in the graphs (in Fig. 26.2 there is the length of the undeformed squad) The condition for the minimum potential energy has the form

In accordance with t (22.4), condition (26.2) is equivalent to the fact that

(in the case where U is a function of only one variable, ). Thus, the position corresponding to the minimum potential energy has the property that the force acting on the body is zero.

In the case shown in Fig. 26.1, conditions (26.2) and (26.3) are also satisfied for x equal to (i.e., for the maximum of U). The position of the ball determined by this value will also be equilibrium. However, this equilibrium, unlike the equilibrium at, will be unstable: it is enough to slightly remove the ball from this position and a force will arise that will move the ball away from the position . The forces that arise when the ball is displaced from a stable equilibrium position (for which ) are directed in such a way that they tend to return the ball to the equilibrium position.

Knowing the type of t function that expresses potential energy, we can make a number of conclusions about the nature of the particle’s movement. Let us explain this using the graph shown in Fig. 26.1, b. If the total energy has the value indicated in the figure, then the particle can move either in the range from to or in the range from to infinity. The particle cannot penetrate into the region, since the potential energy cannot become greater than the total energy (if this happened, the kinetic energy would become negative). Thus, the region represents a potential barrier through which a particle cannot penetrate given a given amount of total energy. The area is called a potential well.

If a particle cannot move away to infinity during its motion, the motion is called finite. If the particle can go as far as desired, the motion is called infinite. A particle in a potential well undergoes finite motion. The motion of a particle with negative full energy in the central field of attractive forces (it is assumed that potential energy vanishes at infinity).

It is known that for the equilibrium of a system with ideal connections it is necessary and sufficient that or. (7)

Since the variations of generalized coordinates are independent of each other and, in general, are not equal to zero, it is necessary that
,
,…,
.

For the equilibrium of a system with holonomic restraining, stationary, ideal constraints, it is necessary and sufficient that all generalized forces corresponding to the selected generalized coordinates be equal to zero.

Case of potential forces:

If the system is in a potential force field, then

,
,…,

,
,…,

That is, the equilibrium positions of the system can only be for those values ​​of generalized coordinates for which the force function U and potential energy P have extreme values ​​( max or min).

The concept of equilibrium stability.

Having determined the positions in which the system can be in equilibrium, it is possible to determine which of these positions are realizable and which are unrealizable, that is, determine which position is stable and which is unstable.

In general, necessary sign of equilibrium stability according to Lyapunov can be formulated as follows:

Let us remove the system from the equilibrium position by providing small modulus values ​​of the generalized coordinates and their velocities. If, upon further consideration of the system, the generalized coordinates and their velocities remain small in magnitude, that is, the system does not deviate far from the equilibrium position, then such an equilibrium position is stable.

Sufficient condition for equilibrium stability system is determined Lagrange-Dirichlet theorem :

If in the equilibrium position of a mechanical system with ideal connections the potential energy has a minimum value, then such an equilibrium position is stable.

,
- sustainable.

Let us present equations (16) from § 107 and (35) or (38) in the form:

Let us show that from these equations, which are consequences of the laws set forth in § 74, all the initial results of statics are obtained.

1. If a mechanical system is at rest, then the velocities of all its points are equal to zero and, therefore, where O is any point. Then equations (40) give:

Thus, conditions (40) are necessary conditions for the equilibrium of any mechanical system. This result contains, in particular, the principle of solidification formulated in § 2.

But for any system, conditions (40) are obviously not sufficient equilibrium conditions. For example, if shown in Fig. 274 points are free, then under the influence of forces they can move towards each other, although conditions (40) for these forces will be satisfied.

Necessary and sufficient conditions for the equilibrium of a mechanical system will be presented in § 139 and 144.

2. Let us prove that conditions (40) are not only necessary, but also sufficient equilibrium conditions for forces acting on an absolutely rigid body. Let a free rigid body at rest begin to be acted upon by a system of forces that satisfies conditions (40), where O is any point, i.e., in particular, point C. Then equations (40) give , and since the body is initially was at rest, then At point C is motionless and the body can only rotate with angular velocity c around a certain instantaneous axis (see § 60). Then, according to formula (33), the body will have . But there is a projection of the vector onto the axis, and since then and from where it follows that and i.e. that when conditions (40) are met, the body remains at rest.

3. From the previous results, in particular, the initial provisions 1 and 2, formulated in § 2, follow, since it is obvious that the two forces depicted in Fig. 2, satisfy conditions (40) and are balanced, and that if we add (or subtract from them) a balanced system of forces to the forces acting on the body, i.e., satisfying conditions (40), then neither these conditions nor equations (40), determining the movement of the body will not change.