Let me consider a material point whose movement 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. 26.1, a).
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 increase only due to a decrease in amplitude 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 total energy in the central field of attractive forces will also be finite (it is assumed that the potential energy vanishes at infinity).
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.
Wikimedia Foundation. 2010.
See what “Mechanical equilibrium” is in other dictionaries:
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
- ... Wikipedia
Phase transitions Article I ... Wikipedia
The state of a thermodynamic system to which it spontaneously comes after a sufficiently long period of time under conditions of isolation from the environment, after which the parameters of the system’s state no longer change over time. Isolation... ... Great Soviet Encyclopedia
EQUILIBRIUM- (1) the mechanical state of immobility of a body, which is a consequence of the R. forces acting on it (when the sum of all forces acting on the body is equal to zero, that is, it does not impart acceleration). R. are distinguished: a) stable, when when deviating from ... ... Big Polytechnic Encyclopedia
Mechanical condition system, in which all its points are motionless with respect to the given reference system. If this reference system is inertial, then R.M. is called. absolute, otherwise relative. Depending on the behavior of the body after... Big Encyclopedic Polytechnic Dictionary
Thermodynamic equilibrium is the state of an isolated thermodynamic system, in which at each point for all chemical, diffusion, nuclear, and other processes, the rate of the forward reaction is equal to the rate of the reverse one. Thermodynamic... ... Wikipedia
Equilibrium- the most probable macrostate of a substance, when variables, regardless of choice, remain constant with a complete description of the system. Equilibrium is distinguished: mechanical, thermodynamic, chemical, phase, etc.: Look... ... Encyclopedic Dictionary of Metallurgy
Contents 1 Classical definition 2 Definition through the energy of the system 3 Types of equilibrium ... Wikipedia
Phase transitions The article is part of the Thermodynamics series. Concept of phase Phase equilibrium Quantum phase transition Sections of thermodynamics Principles of thermodynamics Equation of state ... Wikipedia
This lecture covers the following issues:
1. Conditions for equilibrium of mechanical systems.
2. Stability of balance.
3. An example of determining equilibrium positions and studying 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”, to solve problems in the disciplines “Theory of Machines and Mechanisms” and “Strength of Materials”.
An important case of motion of mechanical systems is their oscillatory motion. Oscillations are repeated movements of a mechanical system relative to some of its positions, occurring more or less regularly over time. The course work examines the oscillatory motion of a mechanical system relative to an equilibrium position (relative or absolute).
A mechanical system can oscillate 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 equilibrium positions and study 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, restraining and holonomic constraints are imposed to be in equilibrium, it is necessary and sufficient that all generalized forces in this system be equal to zero:
Where - generalized force corresponding j- oh generalized coordinate;
s- the number of generalized coordinates in the mechanical system.
If differential equations of motion have been compiled for the system under study in the form of Lagrange equations of the second kind, then to determine possible equilibrium positions it is sufficient to equate the generalized forces to zero and solve the resulting equations with respect to the 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 determined by the above formulas can be realized practically. Depending on the behavior of the system when it deviates from the equilibrium position, one speaks of stability or instability of this position.
Equilibrium stability
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 look at this definition.
To simplify the calculations, we will further agree on 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 numbercan you find another number , that in the case when the initial values of generalized coordinates and velocities will not exceed:
the values of generalized coordinates and velocities during 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 it is always possible to find such sufficiently small initial values, at which the movement 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 clearly depicted in the phase plane (Fig. 1).For a stable equilibrium position, the movement of the representing point, starting in the region [ ] , will not go beyond the region 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 conditions for the stability of an equilibrium position is a rather complex task, so we will limit ourselves to the simplest case: studying the stability of the equilibrium of conservative systems.
Sufficient conditions for the stability of equilibrium positions for such systems are determined 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
P(q)=0.
Functions that have a constant sign and are equal to zero only when all their arguments are zero are called definite. Consequently, in order for the equilibrium position of a 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 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 series expansion of potential energy or from the values of the second derivatives of potential energy with respect to generalized coordinates at the equilibrium position:
From formula (4) it follows that the generalized stiffness coefficients are symmetrical with respect to the indices
For that In order for 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 , which gives necessary and sufficient conditions for the positive definiteness of quadratic forms: quadratic form (3) will be positive definite if the determinant composed of its coefficients and all its principal diagonal minors are positive, i.e. if the odds 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, it is possible to study the positions of relative equilibrium if, instead of potential energy, we introduce into consideration the potential energy of the reduced system.
P An example of determining equilibrium positions and studying their stability
Fig.2
Consider a mechanical system consisting of a tube AB, which is the rod OO 1 connected to the horizontal axis of rotation, and a ball that moves along the tube without friction and is connected to a point A tubes with a spring (Fig. 2). Let us determine the equilibrium positions of the system and evaluate their stability under the following parameters: tube length l 2 = 1 m , rod length l 1 = 0,5 m . undeformed spring length l 0 = 0.6 m spring stiffness c= 100 N/m. Tube weight m 2 = 2 kg, rod - m 1 = 1 kg and the ball - m 3 = 0.5 kg. Distance O.A. equals l 3 = 0.4 m.
Let us write down an expression for the potential energy of the system under consideration. It consists of the potential energy of three bodies located in a uniform field of gravity 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 and the height of its center of gravity above the plane in which the potential energy is considered equal to zero. Let the potential energy be zero in the plane passing through the axis of rotation of the rod O.O. 1, then for gravity
For the elastic force, the potential energy is determined by the magnitude of the deformation
Let us find possible equilibrium positions of the system. The coordinate values at the equilibrium positions are the roots of the following system of equations.
A similar system of equations can be compiled for any mechanical system with two degrees of freedom. In some cases, it is possible to obtain an exact solution of the system. For system (5) such a solution does not exist, so 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 will find all second derivatives of the potential energy with respect to generalized coordinates and from them we will determine the generalized rigidity coefficients.
DEFINITION
Stable balance- this is an equilibrium in which a body, removed from a position of equilibrium and left to itself, returns to its previous position.
This occurs if, with a slight displacement of the body in any direction from the original position, the resultant of the forces acting on the body becomes non-zero 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 a body, taken out of an equilibrium position and left to itself, will deviate even more from the equilibrium position.
In this case, with a slight displacement of the body from the equilibrium position, the resultant of the forces applied to it is non-zero and directed from the equilibrium position. An example is a ball located at the top point of a convex spherical surface (Fig. 1 b).
DEFINITION
Indifferent Equilibrium- this is an equilibrium in which a body, taken out of an equilibrium position and left to its own devices, does not change its position (state).
In this case, with small displacements of the body from the original 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. Different types of body balance on a support: a) stable balance; b) unstable equilibrium; c) indifferent equilibrium.
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 equilibrium.
DEFINITION
Static balance- this is an equilibrium when, under the influence of applied forces, the body is at rest.
Dynamic balance- this is an equilibrium when, due to the action of forces, the body does not change its movement.
A lantern suspended on cables, or any building structure, is in a state of static equilibrium. As an example of dynamic equilibrium, consider a wheel that rolls on a flat surface in the absence of friction forces.
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.
