This lecture addresses the following issues:

1. Equilibrium conditions for mechanical systems.

2. Stability of balance.

3. An example of the determination of equilibrium positions and the study of their stability.

The study of these issues is necessary to study the oscillatory movements of a mechanical system relative to the equilibrium position in the discipline "Machine parts", for solving problems in the disciplines "Theory of machines and mechanisms" and "Resistance of materials".

An important case of motion of mechanical systems is their oscillatory motion. Oscillations are repetitive movements of a mechanical system relative to some of its position, occurring more or less regularly in time. The course work examines the oscillatory motion of a mechanical system relative to the equilibrium position (relative or absolute).

A mechanical system can vibrate for a sufficiently long period of time only near a stable equilibrium position. Therefore, before composing the equations of oscillatory motion, it is necessary to find the equilibrium positions and investigate their stability.

Equilibrium conditions for mechanical systems.

According to the principle of possible displacements (the basic equation of statics), in order for a mechanical system, on which ideal, stationary, holding and holonomic constraints are imposed, to be in equilibrium, it is necessary and sufficient that all generalized forces are equal to zero in this system:

where - generalized force corresponding j - th generalized coordinate;

s- the number of generalized coordinates in the mechanical system.

If for the system under study, differential equations of motion were compiled in the form of Lagrange equations of the second kind, then to determine the possible equilibrium positions, it is sufficient to equate the generalized forces to zero and solve the resulting equations with respect to generalized coordinates.

If the mechanical system is in equilibrium in a potential force field, then from equations (1) we obtain the following equilibrium conditions:

Therefore, in the equilibrium position, the potential energy has an extreme value. Not every equilibrium defined by the above formulas can be realized in practice. Depending on the behavior of the system when deviating from the equilibrium position, one speaks of the stability or instability of this position.

Stable balance

The definition of the concept of stability of an equilibrium position was given at the end of the 19th century in the works of the Russian scientist A.M. Lyapunov. Let's consider this definition.

To simplify the calculations, we will further agree on the generalized coordinates q 1 , q 2 ,...,q s count from the equilibrium position of the system:

where

An equilibrium position is called stable if for any arbitrarily small numberyou can find such a different number , that in the case when the initial values ​​of the generalized coordinates and velocities will not exceed:

the values ​​of the generalized coordinates and velocities during the further movement of the system will not exceed .

In other words, the equilibrium position of the system q 1 = q 2 = ...= q s = 0 is called sustainable if one can always find such sufficiently small initial valuesat which the motion of the systemwill not leave any given arbitrarily small neighborhood of the equilibrium position... For a system with one degree of freedom, the stable motion of the system can be graphically depicted in the phase plane (Fig. 1).For a stable equilibrium position, the movement of the representing point, starting in the area [ ] , will not go beyond the area in the future.

Fig. 1

The equilibrium position is called asymptotically stable , if over time the system approaches the equilibrium position, that is

Determining the stability conditions for an equilibrium position is a rather complicated problem, therefore, we restrict ourselves to the simplest case: the study of the equilibrium stability of conservative systems.

Sufficient conditions for the stability of equilibrium positions for such systems are determined the Lagrange - Dirichlet theorem : the equilibrium position of a conservative mechanical system is stable if, in the equilibrium position, the potential energy of the system has an isolated minimum .

The potential energy of a mechanical system is determined to within a constant. Let us choose this constant so that in the equilibrium position the potential energy is equal to zero:

P (0) = 0.

Then, for a system with one degree of freedom, a sufficient condition for the existence of an isolated minimum, along with the necessary condition (2), will be the condition

Since in the equilibrium position the potential energy has an isolated minimum and P (0) = 0 , then in some finite neighborhood of this position

П (q) = 0.

Functions that have a constant sign and are equal to zero only for zero values ​​of all their arguments are called definite... Consequently, in order for the equilibrium position of the mechanical system to be stable, it is necessary and sufficient that in the vicinity of this position the potential energy is a positive definite function of the generalized coordinates.

For linear systems and for systems that can be reduced to linear for small deviations from the equilibrium position (linearized), the potential energy can be represented in the form of a quadratic form of generalized coordinates

where - generalized stiffness coefficients.

Generalized coefficientsare constant numbers that can be determined directly from the expansion of the potential energy in a series or from the values ​​of the second derivatives of the potential energy with respect to generalized coordinates in the equilibrium position:

It follows from formula (4) that the generalized stiffness coefficients are symmetric with respect to the indices

For In order for the sufficient conditions for the stability of the equilibrium position to be satisfied, the potential energy must be a positive definite quadratic form of its generalized coordinates.

In mathematics, there is Sylvester criterion giving necessary and sufficient conditions for the positive definiteness of quadratic forms: the quadratic form (3) will be positive definite if the determinant composed of its coefficients and all its main diagonal minors are positive, i.e. if the coefficients will satisfy the conditions

.....

In particular, for a linear system with two degrees of freedom, the potential energy and the conditions of the Sylvester criterion will have the form

In a similar way, one can study the positions of relative equilibrium if, instead of the potential energy, the potential energy of the reduced system is introduced.

NS An example of determining equilibrium positions and studying their stability

Fig. 2

Consider a mechanical system consisting of a tube AB, which is the pivot OO 1 connected to the horizontal axis of rotation, and a ball that moves through the tube without friction and is connected to a point A tube with a spring (Fig. 2). Let us determine the equilibrium positions of the system and estimate their stability for the following parameters: tube length l 2 = 1 m , rod length l 1 = 0,5 m . length of undeformed spring l 0 = 0.6 m, spring rate c= 100 N / m. Tube weight m 2 = 2 kg, rods - m 1 = 1 kg and the ball - m 3 = 0.5 kg. Distance OA equals l 3 = 0.4 m.

Let us write down the expression for the potential energy of the system under consideration. It consists of the potential energy of three bodies in a uniform gravity field and the potential energy of a deformed spring.

The potential energy of a body in a gravity field is equal to the product of the body's weight by the height of its center of gravity above the plane in which the potential energy is considered to be zero. Let the potential energy be zero in the plane passing through the axis of rotation of the rod OO 1, then for the forces of gravity

For the elastic force, the potential energy is determined by the amount of deformation

Let us find the possible equilibrium positions of the system. The values ​​of the coordinates at the equilibrium positions are the roots of the following system of equations.

A similar system of equations can be drawn up for any mechanical system with two degrees of freedom. In some cases, an exact solution to the system can be obtained. For system (5), such a solution does not exist; therefore, the roots must be sought using numerical methods.

Solving the system of transcendental equations (5), we obtain two possible equilibrium positions:

To assess the stability of the obtained equilibrium positions, we find all the second derivatives of the potential energy with respect to the generalized coordinates, and from them we determine the generalized stiffness coefficients.

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

To find the equilibrium conditions for an absolutely rigid body, it is necessary to mentally break it down into a large number of sufficiently small elements, each of which can be represented by a material point. All these elements interact with each other - these forces of interaction 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 stationary point to be zero), the geometric sum of the forces acting on this point must be equal to zero. If the body is at rest, then all its points (elements) are also at rest. Therefore, for any point of the body, you 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 the balance of a body it is necessary and sufficient that the geometric sum of all forces acting on any element of this body is equal to zero.

From it is easy to obtain the first condition for the balance of a body (system of bodies). To do this, it is enough to sum up the equation over all the 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 equilibrium of a rigid body(body systems) is the equality to zero of the geometric sum of all external forces applied to the body.

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

The second condition for 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 for a rigid body in the case of an arbitrary number of external forces are as follows:

.

Equilibrium is a state of the system in which the forces acting on the system are balanced with each other. Equilibrium can be stable, unstable, or indifferent.

The concept of equilibrium is one of the most universal in the natural sciences. It applies to any system, be it a system of planets moving in stationary orbits around a star, or a population of tropical fish in an atoll lagoon. But the easiest way to understand the concept of the equilibrium state of a system is on the example of mechanical systems. In mechanics, it is considered that a system is in equilibrium if all forces acting on it are completely balanced with each other, that is, they extinguish each other. If you are reading this book, for example, while sitting in a chair, then you are in a state of balance, since the force of gravity pulling you down is completely compensated by the force of the chair's pressure on your body, acting from the bottom up. You do not fall or take off precisely because you are in a state of balance.

There are three types of balance, corresponding to three physical situations.

Stable balance

This is what most people usually understand by "balance". Imagine a ball at the bottom of a spherical bowl. At rest, it is located strictly in the center of the bowl, where the action of the force of the Earth's gravitational attraction is balanced by the reaction force of the support directed strictly upward, and the ball rests there just like you are resting in your chair. If you move the ball away from the center, rolling it to the side and up towards the edge of the bowl, then, as soon as you release it, it immediately rushes back to the deepest point in the center of the bowl - in the direction of a stable equilibrium position.

Sitting in a chair, you are at rest due to the fact that the system, consisting of your body and the chair, is in a state of stable equilibrium. Therefore, when you change some parameters of this system - for example, when your weight increases, if, say, a child sits on your knees - the chair, being a material object, will change its configuration in such a way that the reaction force of the support increases, and you will remain in a position of stable balance (the most that can happen is the pillow under you will rinse a little deeper).

In nature, there are many examples of stable equilibrium in various systems (and not only mechanical ones). Consider, for example, the predator-prey relationship in an ecosystem. The ratio of the numbers of closed populations of predators and their prey quickly enough comes to an equilibrium state - so many hares in the forest from year to year consistently fall on so many foxes, relatively speaking. If, for some reason, the number of prey populations changes sharply (due to a surge in the birth rate of hares, for example), the ecological balance will very soon be restored due to a rapid increase in the number of predators, which will begin to exterminate hares at an accelerated rate until the number of hares is brought back to normal and will not begin to die out of hunger themselves, bringing their own livestock back to normal, as a result of which the populations of both hares and foxes will return to the norm that was observed before the burst of birth rates among hares. That is, in a stable ecosystem, internal forces also act (although not in the physical sense of the word), striving to return the system to a state of stable equilibrium in the event that the system deviates from it.

Similar effects can be observed in economic systems. A sharp drop in the price of a product leads to a surge in demand from hunters for cheapness, a subsequent reduction in inventories and, as a consequence, an increase in prices and a drop in demand for the product - and so on until the system returns to a state of stable price equilibrium of supply and demand. (Naturally, in real systems, both ecological and economic, external factors can act that deviate the system from an equilibrium state - for example, the seasonal shooting of foxes and / or hares or government price regulation and / or consumption quotas. equilibrium, an analogue of which in mechanics would be, for example, deformation or tilt of the bowl.)

Unstable equilibrium

Not every equilibrium, however, is stable. Imagine a ball balancing on a knife blade. The force of gravity directed strictly downward in this case, obviously, is also completely balanced by the force directed upward by the reaction force of the support. But as soon as the center of the ball is deflected away from the rest point, which falls on the line of the blade at least a fraction of a millimeter (and for this, a meager force of force is enough), the balance will be instantly disturbed and the force of gravity will begin to drag the ball further and further from it.

An example of an unstable natural equilibrium is the Earth's heat balance when the periods of global warming change with new ice ages and vice versa ( cm. Milankovitch cycles). The average annual temperature of the surface of our planet is determined by the energy balance between the total solar radiation reaching the surface and the total thermal radiation of the Earth into outer space. This heat balance becomes unstable as follows. Some winter there is more snow than usual. The next summer, the heat is not enough to melt the excess snow, and the summer is also colder than usual due to the fact that due to the excess of snow, the Earth's surface reflects back into space a greater proportion of the sun's rays than before. Because of this, the next winter turns out to be even more snowy and colder than the previous one, and in the following summer, even more snow and ice remains on the surface, reflecting solar energy into space ... It is easy to see that the more such a global climate system deviates from the initial point of thermal equilibrium, the faster the processes grow, leading the climate further away from it. Ultimately, many kilometers of glaciers are formed on the Earth's surface in the circumpolar regions for many years of global cooling, which are inexorably moving towards ever lower latitudes, bringing with them another ice age to the planet. So it is difficult to imagine a more precarious balance than the global-climatic one.

A type of unstable equilibrium deserves special mention, called metastable, or quasi-stable equilibrium. Imagine a ball in a narrow, shallow groove — for example, on a curved skate blade with its tip up. A slight - by a millimeter or two - deviation from the equilibrium point will lead to the emergence of forces that return the ball to an equilibrium state in the center of the groove. However, a little more force is enough to bring the ball out of the metastable equilibrium zone, and it will fall off the skate blade. Metastable systems, as a rule, have the property of staying for some time in a state of equilibrium, after which they "break" from it as a result of any fluctuations of external influences and "fall" into an irreversible process characteristic of unstable systems.

A typical example of quasi-stable equilibrium is observed in the atoms of the working substance of some types of laser installations. The electrons in the atoms of the working fluid of the laser occupy metastable atomic orbits and remain on them until the flight of the very first light quantum, which "knocks" them from the metastable orbit to a lower stable one, while emitting a new quantum of light, coherent to the flying one, which, in turn, knocks down the electron of the next atom from the metastable orbit, etc. As a result, an avalanche-like reaction of the emission of coherent photons that form a laser beam is triggered, which, in fact, underlies the action of any laser.

Indifferent balance

An intermediate case between stable and unstable equilibrium is the so-called indifferent equilibrium, in which any point of the system is an equilibrium point, and the deviation of the system from the initial point of rest does not change anything in the alignment of forces within it. Imagine a ball on a perfectly smooth horizontal table - wherever you move it, it will remain in equilibrium.

DEFINITION

Stable balance- this is a balance in which the body, taken out of a position of balance and left to itself, returns to its previous position.

This happens if, with a slight displacement of the body in any direction from the initial position, the resultant of the forces acting on the body becomes nonzero and is directed towards the equilibrium position. For example, a ball lying at the bottom of a spherical depression (Fig. 1 a).

DEFINITION

Unstable equilibrium- this is an equilibrium in which the body, taken out of the equilibrium position and left to itself, will deviate even more from the equilibrium position.

In this case, with a small displacement of the body from the equilibrium position, the resultant of the forces applied to it is nonzero and is directed from the equilibrium position. An example is a ball located at the top of a convex spherical surface (Fig. 1 b).

DEFINITION

Indifferent balance- this is a balance in which the body, taken out of a position of balance and left to itself, does not change its position (state).

In this case, at small displacements of the body from the initial position, the resultant of the forces applied to the body remains equal to zero. For example, a ball lying on a flat surface (Fig. 1, c).

Fig. 1. Various types of body balance on a support: a) stable balance; b) unstable balance; c) indifferent balance.

Static and dynamic balance of bodies

If, as a result of the action of forces, the body does not receive acceleration, it can be at rest or move uniformly in a straight line. Therefore, we can talk about static and dynamic balance.

DEFINITION

Static balance- this is such a balance when, under the action of the applied forces, the body is at rest.

Dynamic balance- this is such a balance when, under the action of forces, the body does not change its motion.

In a state of static equilibrium there is a lantern suspended on cables, any building structure. As an example of dynamic equilibrium, we can consider a wheel that rolls on a flat surface in the absence of frictional forces.

A body is at rest (or moves uniformly and rectilinearly) if the vector sum of all the forces acting on it is zero. The forces are said to counterbalance each other. When we are dealing with a body of a certain geometric shape, when calculating the resultant force, all forces can be applied to the center of mass of the body.

Equilibrium condition for bodies

In order for a body that does not rotate to be in equilibrium, it is necessary that the resultant of all forces acting on it be equal to zero.

F → = F 1 → + F 2 → +. ... + F n → = 0.

The figure above shows the equilibrium of a rigid body. The bar is in a state of equilibrium under the influence of three forces acting on it. The lines of action of the forces F 1 → and F 2 → intersect at the point O. The point of application of the force of gravity is the center of mass of the body C. These points lie on one straight line, and when calculating the resultant force F 1 →, F 2 → and m g → are reduced to point C.

The condition of equality to zero of the resultant of all forces is not enough if the body can rotate around some axis.

The shoulder of the force d is the length of the perpendicular drawn from the line of action of the force to the point of its application. The moment of force M is the product of the shoulder of the force by its modulus.

The moment of force tends to rotate the body around the axis. Those moments that rotate the body counterclockwise are considered positive. The unit of measurement of the moment of force in the international system SI is 1 Nyutonmetr.

Definition. Rule of the Moments

If the algebraic sum of all the moments applied to the body relative to the fixed axis of rotation is zero, then the body is in equilibrium.

M 1 + M 2 +. ... + M n = 0

Important!

In the general case, for the balance of bodies, two conditions must be met: equality to zero of the resultant force and observance of the rule of moments.

There are different types of balance in mechanics. So, they distinguish between stable and unstable, as well as indifferent equilibrium.

A typical example of indifferent equilibrium is a rolling wheel (or ball), which, if stopped at any point, will be in a state of equilibrium.

Stable equilibrium is such a balance of a body when, with its small deviations, forces or moments of forces arise that tend to return the body to an equilibrium state.

Unstable equilibrium is a state of equilibrium, with a small deviation from which the forces and moments of forces tend to unbalance the body even more.

In the picture above, the position of the ball (1) is an indifferent equilibrium, (2) is an unstable equilibrium, (3) is a stable equilibrium.

A body with a fixed axis of rotation can be in any of the described equilibrium positions. If the axis of rotation passes through the center of mass, an indifferent equilibrium arises. In a stable and unstable equilibrium, the center of mass is located on a vertical line that passes through the axis of rotation. When the center of mass is below the axis of rotation, equilibrium is stable. Otherwise, the opposite is true.

A special case of balance is the balance of the body on a support. In this case, the elastic force is distributed over the entire base of the body, and does not pass through one point. The body is at rest in equilibrium when a vertical line drawn through the center of mass crosses the support area. Otherwise, if the line from the center of mass does not fall into the contour formed by the lines connecting the pivot points, the body overturns.

An example of body balance on a support is the famous Leaning Tower of Pisa. According to legend, Galileo Galilei dropped balls from it when he conducted his experiments to study the free fall of bodies.

A line drawn from the center of mass of the tower intersects the base approximately 2.3 m from its center.

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