As you know, a body that is not limited in any way in its movements is called free, since it can move in any direction. From here, every free solid has six degrees of freedom of movement. It has the ability to produce the following movements: three translational movements, corresponding to three main coordinate systems, and three rotational movements around these three coordinate axes.
Imposing connections (fixing) reduces the number of degrees of freedom. Thus, if a body is fixed at one point, it cannot move along the coordinate axes; its movements are limited only to rotation around these axes, i.e. the body has three degrees of freedom. In the case when two points are fixed, the body has only one degree of freedom; it can only rotate around a line (axis) passing through both of these points. And finally, with three fixed points that do not lie on the same line, the number of degrees of freedom is zero, and no body movements can occur. In humans, the passive apparatus of movement consists of parts of his body called links. They are all connected to each other, so they lose the ability to perform three types of movements along the coordinate axes. They only have the ability to rotate around these axes. Thus, the maximum number of degrees of freedom that one body link can have in relation to another link adjacent to it is three.
This applies to the most mobile joints human body, having a spherical shape.
Sequential or branched connections of body parts (links) form kinematic chains.
In humans there are:
- - open kinematic chains having a free movable end, fixed only at one end (for example, an arm in relation to the body);
- - closed kinematic chains, fixed at both ends (for example, vertebra - rib - sternum - rib - vertebra).
It should be noted that this concerns the potential range of movements in the joints. In reality, in a living person, these indicators are always lower, which has been proven by numerous works of domestic researchers - P. F. Lesgaft, M. F. Ivanitsky, M. G. Prives, N. G. Ozolin, etc. On the amount of mobility in bone joints in a living person is influenced by a number of factors related to age, gender, individual characteristics, functional state nervous system, degree of muscle strain, temperature environment, time of day and, finally, what is important for athletes, the degree of training. Thus, in all bone connections (discontinuous and continuous), the degree of mobility in young people is greater than in older people; On average, women have more than men. The amount of mobility is influenced by the degree of stretching of those muscles that are on the side opposite to movement, as well as the strength of the muscles producing this movement. The more elastic the first of these muscles and the stronger the second, the greater the range of movements in a given bone connection, and vice versa. It is known that in a cold room movements have a smaller scope than in a warm room; in the morning they are less than in the evening. The use of different exercises has different effects on joint mobility. Thus, systematic training with “flexibility” exercises increases the range of motion in the joints, while “strength” exercises, on the contrary, reduce it, leading to “stiffening” of the joints. However, a decrease in the range of motion in joints when using strength exercises is not absolutely inevitable. It can be prevented by the right combination of strength training and stretching exercises for the same muscle groups.
In the open kinematic chains of the human body, mobility is calculated in tens of degrees of freedom. For example, the mobility of the wrist relative to the scapula and the mobility of the tarsus relative to the pelvis have seven degrees of freedom, and the tips of the fingers of the hand relative to the chest have 16 degrees of freedom. If we sum up all the degrees of freedom of the limbs and head relative to the body, then this will be expressed by the number 105, composed of the following positions:
- - head - 3 degrees of freedom;
- - arms - 14 degrees of freedom;
- - legs - 12 degrees of freedom;
- - hands and feet - 76 degrees of freedom.
For comparison, we point out that the vast majority of machines have only one degree of freedom of movement.
In ball and socket joints, rotations about three mutually perpendicular axes are possible. The total number of axes around which rotations are possible in these joints is infinitely large. Consequently, with regard to spherical joints, we can say that the links articulated in them, out of possible six degrees of freedom of movement, have three degrees of freedom and three degrees of coupling.
Joints with two degrees of freedom of movement and four degrees of coupling have less mobility. These include joints of ovoid or elliptical and saddle shapes, i.e. biaxial. They allow movements around these two axes.
The body links in those joints that have one axis of rotation, i.e., have one degree of freedom of mobility and at the same time five degrees of connectivity. have two fixed points.
The majority of joints in the human body have two or three degrees of freedom. With several degrees of freedom of movement (two or more), an infinite number of trajectories are possible. The connections of the skull bones have six degrees of connection and are immobile. The connection of bones with the help of cartilage and ligaments (synchondrosis and syndesmosis) can in some cases have significant mobility, which depends on the elasticity and on the size of the cartilaginous or connective tissue formations located between these bones.
Systems with two degrees of freedom are a special case of systems with several degrees of freedom. But these systems are the simplest, allowing one to obtain in final form calculation formulas for determining vibration frequencies, amplitudes and dynamic deflections.
yBeam deflections due to inertial forces:
P 2 =1 
(1)
The signs (-) in expressions (1) are due to the fact that inertial forces and units. the movements are in the opposite direction.
We believe that mass vibrations occur according to the harmonic law:
(2)
Let's find the acceleration of mass motion:
(3)
Substituting expressions (2) and (3) into equation (1) we obtain:
(5)
We consider the amplitudes of oscillations A 1 and A 2 unknown, and we transform the equations:
(6)
The solution to the system of homogeneous equations A 1 = A 2 =0 does not suit us; in order to obtain a non-zero solution, we equate the determinants of the system (6) to zero:
(7)
Let us transform equation (8), considering the circular frequency of natural oscillations unknown:
Equation (9) is called the biharmonic equation of free oscillations of systems with two degrees of freedom.
Replacing the variable 2 =Z, we get
from here we determine Z 1 and Z 2.
As a result, the following conclusions can be drawn:
1. Free vibrations of systems with two degrees of freedom occur with two frequencies 1 and 2. The lower frequency 1 is called the fundamental or fundamental tone, the higher frequency 2 is called the second frequency or overtone.
Free vibrations of systems with n-degrees of freedom are n-tone, consisting of n-free vibrations.
2. The movements of masses m 1 and m 2 are expressed by the following formulas:
i.e., if oscillations occur with a frequency 1, then at any moment of time the mass movements have the same signs.
If oscillations occur only with a frequency 2, then the mass movements at any time have opposite signs.
With simultaneous oscillations of masses with frequencies 1 and 2, the system mainly oscillates at frequency 1 and an overtone with frequency 2 fits into these oscillations.
If a system with two degrees of freedom is subject to a driving force with frequency , then it is necessary that:
0.7 1 .
Lecture 9
Oscillations of systems with an infinite number of degrees of freedom.
Theory mechanical vibrations has numerous and very diverse applications in almost all areas of technology. Regardless of the purpose and design solution of various mechanical systems, their vibrations are subject to the same physical laws, the study of which is the subject of the theory of vibrations of elastic systems. The linear theory of oscillations has been most fully developed. The theory of oscillations of systems with several degrees of freedom was given back in the 18th century by Lagrange in his classic work “Analytical Mechanics”.
Joseph Louis Lagrange (1736 - 1813) - professor of mathematics in Turin from the age of 19. Since 1759 - member, and since 1766 - president of the Berlin Academy of Sciences; from 1787 he lived in Paris. In 1776 he was elected an honorary foreign member of the St. Petersburg Academy of Sciences.
IN late XIX century, Rayleigh laid the foundations of the linear theory of oscillations of systems with an infinite degree of degrees of freedom (i.e., with a continuous distribution of mass throughout the entire volume of the deformable system). In the 20th century, the linear theory could be said to have been completed (the Bubnov-Galerkin method, which also makes it possible to determine higher oscillation frequencies using successive approximations).
John William Strett (Lord Rayleigh) (1842 - 1919) - English physicist, author of a number of works on the theory of oscillations.
Ivan Grigorievich Bubnov (1872 - 1919) - one of the founders of ship structural mechanics. Professor at the St. Petersburg Polytechnic Institute, since 1910 - at the Maritime Academy.
Boris Grigorievich Galerkin (1871-1945) - professor at the Leningrad Polytechnic Institute.
Rayleigh's formula is most popular in the theory of vibrations and stability of elastic systems. The idea underlying the derivation of Rayleigh's formula comes down to the following. During monoharmonic (one-tone) free oscillations of an elastic system with frequency , the movements of its points occur in time according to the harmonic law:
where 1 (x,y,z), 2 (x,y,z), 3 (x,y,z) are functions of the spatial coordinates of the point that determine the vibration form under consideration (amplitude).
If these functions are known, then the frequency of free vibrations can be found from the condition that the sum of the kinetic and potential energy of the body is constant. This condition leads to an equation containing only one unknown quantity.
However, these functions are not known in advance. The guiding idea of the Rayleigh method is to specify these functions, matching their choice with the boundary conditions and the expected shape of the vibrations.
Let's consider in more detail the implementation of this idea for plane bending vibrations of a rod; the shape of the vibrations is described by the function =(x). Free oscillations are described by the dependence
potential energy of a bent rod
(2)
kinetic energy
(3)
Where l- length of the rod, m=m(x) intensity of the distributed mass of the rod;
Curvature of the curved axis of the rod; - speed of transverse vibrations.
Given (1)
.
(4)
(5)
Over time, each of these quantities changes continuously, but, according to the law of conservation of energy, their sum remains constant, i.e.
or by substituting expressions (4), (5) here
(7)
This leads to Rayleigh's formula:
(8)
If concentrated loads with masses M i are associated with a rod with a distributed mass m, then Rayleigh’s formula takes the form:
(9)
The entire course of the derivation shows that, within the framework of the accepted assumptions (the validity of the technical theory of bending of rods, the absence of inelastic resistance), this formula is accurate if (x) is the true form of vibrations. However, the function(x) is unknown in advance. The practical significance of Rayleigh's formula is that it can be used to find the natural frequency, given the vibration shape(x). At the same time, a more or less serious element of proximity is introduced into the decision. For this reason, Rayleigh's formula is sometimes called an approximate formula.
m=cosnt Let us take as the vibration form the function:(x)=ax 2, which satisfies the kinematic boundary conditions of the problem.
We define:
According to formula (8)
This result differs significantly from the exact one
More accurate is the Grammel formula, which has not yet become as popular as the Rayleigh formula (perhaps due to its relative “youth” - it was proposed in 1939).
Let us again dwell on the same problem of free bending vibrations of a rod.
Let (x) be the specified form of free oscillations of the rod. Then the intensity of the maximum inertial forces is determined by the expression m 2 , where, as before, m=m(x) is the intensity of the distributed mass of the rod; 2 is the square of the natural frequency. These forces reach the specified value at the moment when the deflections are maximum, i.e. are determined by the function(x).
Let us write the expression for the highest potential bending energy in terms of bending moments caused by the maximum inertial forces:
. (10)
Here 
- bending moments caused by load m 2 . Let us denote the bending moment caused by the conditional load m, i.e. 2 times less than the inertial force.
,
(11)
and expression (10) can be written as:
. (12)
Highest kinetic energy, same as above
. (13)
Equating expressions (12) and (13) we arrive at Grammel’s formula:
(14)
To calculate using this formula, you must first specify a suitable function (x). After this, the conditional load m=m(x)(x) is determined and the expressions caused by the conditional load m are written. Using formula (14), the natural oscillation frequency of the system is determined.
Example: (consider the previous one)
y 
m(x)·(x)=max 2
According to (3.7), the system of equations for II =2 has the form:
Since we are talking about free oscillations, the right-hand side of system (3.7) is taken equal to zero.
We are looking for a solution in the form
After substituting (4.23) into (4.22) we get:
This system of equations is valid for an arbitrary t, therefore, expressions enclosed in square brackets are equal to zero. Thus we get linear system algebraic equations relative to L and IN.
An obvious trivial solution to this system L= Oh, B = O according to (4.23) corresponds to the absence of oscillations. However, along with this solution, there is also a non-trivial solution L * O, V F 0 provided that the determinant of the system A ( To 2) equal to zero:
This determinant is called frequency, and the equation is relative k - frequency equation. Expanded function A(k 2) can be represented as
Rice. 4.5
For YatsYad - ^2 > ® and with n ^-4>0 graph A (k 2) has the form of a parabola intersecting the abscissa axis (Fig. 4.5).
Let us show that for oscillations around a stable equilibrium position, the above inequalities are satisfied. Let us transform the expression for kinetic energy as follows:
At q, = 0 we have T = 0,5a.
Next, we prove that the roots of the frequency equation (4.25) are two positive values To 2 and to 2(in the theory of oscillations, a lower index corresponds to a lower frequency, i.e. k ( For this purpose, we first introduce the concept of partial frequency. This term is understood as the natural frequency of a system with one degree of freedom, obtained from the original system by fixing all generalized coordinates except one. So, for example, if in the first of the equations of the system we (4.22) accept q 2 = 0, then the partial frequency will be p ( =yjc u /a n. Similarly, fixing p 2 ~^c n /a 21.
For the frequency equation (4.25) to have two real roots k x And k 2, it is necessary and sufficient that, firstly, the graph of function A (to 2) at k = 0 would have a positive ordinate, and secondly, that it intersect the x-axis. The case of multiple frequencies k ( = k. ), as well as the appeal lowest frequency to zero, is not considered here. The first of these conditions is met, since d (0) = c„c 22 - with and> 0 It is easy to verify the validity of the second condition by substituting the dependence (4.25) k = k = p 2 ; in this case, A(p, 2) Information of this kind in engineering calculations facilitates forecasts and estimates.
The resulting two frequency values To, And to 2 correspond to partial solutions of the form (4.23), therefore general solution has the following form:
Thus, each of the generalized coordinates participates in a complex oscillatory process, which is the addition of harmonic movements with different frequencies, amplitudes and phases (Fig. 4.6). Frequencies k t And to 2 in the general case are incommensurable, therefore q v c, are not periodic functions.
Rice. 4.6
The ratio of the amplitudes of free vibrations at a fixed natural frequency is called the shape coefficient. For a system with two degrees of freedom, the shape coefficients (3.= BJA." are determined directly from equations (4.24):
Thus, the coefficients of the form p, = V 1 /A [ and r.,= V.,/A., depend only on the parameters of the system and do not depend on the initial conditions. Shape coefficients are characterized for the natural frequency under consideration To. distribution of amplitudes along the oscillatory circuit. The combination of these amplitudes forms the so-called vibration form.
A negative value of the form factor means that the oscillations are in antiphase.
When using standard computer programs, they sometimes use normalized shape coefficients. This term means
In the coefficient p' g index i corresponds to the coordinate number, and the index G- frequency number. It's obvious that 
or It is easy to notice that p*
In the system of equations (4.28), the remaining four unknowns A g A 2, oc, cx 2 are determined using the initial conditions:
The presence of a linear resistance force, just as in a system with one degree of freedom, leads to the damping of free oscillations.
Rice. 4.7
Example. Let us determine the natural frequencies, partial frequencies and shape factors for the oscillatory system shown in Fig. 4.7, A. Taking absolute displacements of mass.g as generalized coordinates, = q v x 2 = q. r Let us write down the expressions for the kinetic and potential energies:
Thus,
After substituting into the frequency equations (4.25) we obtain
Moreover, according to (4.29)
In Fig. 4.7, b the vibration modes are given. In the first form of oscillation, the masses move synchronously in one direction, and in the second, in the opposite direction. In addition, in the latter case a cross section appeared N, not participating in the oscillatory process with its own frequency k r This is the so-called vibration unit.
The theory of free oscillations of systems with several degrees of freedom is constructed in a similar way to how one-dimensional oscillations were considered in § 21.
Let the potential energy of the system U, as a function of generalized coordinates, have a minimum at . Introducing small offsets
and expanding U in terms of them up to second-order terms, we obtain the potential energy in the form of a positive definite quadratic form
where we again count the potential energy from its minimum value. Since the coefficients and are included in (23.2) multiplied by the same value, it is clear that they can always be considered symmetrical in their indices
In kinetic energy, which in the general case has the form
(see (5.5)), we put it in the coefficients and, denoting the constants by , we obtain it in the form of a positive definite quadratic form
Thus, the Lagrangian function of a system performing free small oscillations:
Let us now compose the equations of motion. To determine the derivatives included in them, we write the total differential of the Lagrange function
Since the value of the sum does not depend, of course, on the designation of the summation indices, we change in the first and third terms in brackets i by k, and k by i; taking into account the symmetry of the coefficients, we obtain:
From this it is clear that
Therefore Lagrange's equations
(23,5)
They represent a system of linear homogeneous differential equations with constant coefficients.
By general rules solutions to such equations, we look for s unknown functions in the form
where are some, as yet undefined, constants. Substituting (23.6) into system (23.5), we obtain, by reduction to a system of linear homogeneous algebraic equations, which must be satisfied by the constants:
In order for this system to have non-zero solutions, its determinant must vanish
Equation (23.8) - the so-called characteristic equation is an equation of degree s with respect to It has, in the general case, s different real positive roots (in special cases, some of these roots may coincide). The quantities determined in this way are called the natural frequencies of the system.
The reality and positivity of the roots of equation (23.8) are already obvious from physical considerations. Indeed, the presence of an imaginary part in y would mean the presence in the time dependence of the coordinates (23.6) (and with them the velocities) of an exponentially decreasing or exponentially increasing factor. But the presence of such a multiplier in in this case is unacceptable, since it would lead to a change in the total energy of the system over time, contrary to the law of its conservation.
The same thing can be verified purely mathematically. Multiplying equation (23.7) by and then summing by we get:
The quadratic forms in the numerator and denominator of this expression are real due to the reality and symmetry of the coefficients and, indeed,
They are also significantly positive, and therefore positively
After the frequencies have been found, by substituting each of them into equations (23.7), one can find the corresponding values of the coefficients. If all the roots of the characteristic equation are different, then, as is known, the coefficients A are proportional to the minors of the determinant (23.8), in which the replacement We denote these minors with the corresponding value through Do. A particular solution to the system of differential equations (23.5) therefore has the form
where is an arbitrary (complex) constant.
The general solution is given by the sum of all s particular solutions. Moving on to the real part, we write it in the form
where we introduced the notation
(23,10)
Thus, the change in each of the coordinates of the system over time represents the superposition of s simple periodic oscillations with arbitrary amplitudes and phases, but having well-defined frequencies.
The question naturally arises: is it possible to choose generalized coordinates in such a way that each of them performs only one simple oscillation? The very form of the general integral (23.9) indicates the path to solving this problem.
In fact, considering s relations (23.9) as a system of equations with s unknown quantities, we can, having resolved this system, express the quantities through the coordinates. Therefore, quantities can be considered as new generalized coordinates. These coordinates are called normal (or principal), and the simple periodic oscillations they perform are called normal oscillations of the system.
Normal coordinates satisfy, as is clear from their definition, the equations
(23,11)
This means that in normal coordinates the equations of motion break down into s equations independent of each other. The acceleration of each normal coordinate depends only on the value of the same coordinate, and to fully determine its time dependence it is necessary to know the initial values only of itself and its corresponding speed. In other words, the normal oscillations of the system are completely independent.
From the above, it is obvious that the Lagrange function, expressed in terms of normal coordinates, breaks down into a sum of expressions, each of which corresponds to a one-dimensional oscillation with one of the frequencies, i.e., it has the form
(23,12)
where are positive constants. From a mathematical point of view, this means that by transformation (23.9) both quadratic forms - kinetic energy (23.3) and potential energy (23.2) are simultaneously reduced to a diagonal form.
Typically, normal coordinates are chosen so that the coefficients of the squared velocities in the Lagrange function are equal to 1/2. To do this, it is enough to define the normal coordinates (we now denote them) by the equalities
All of the above changes little in the case when among the roots of the characteristic equation there are multiple roots. The general form (23.9), (23.10) of the integral of the equations of motion remains the same (with the same number s of terms) with the only difference that the coefficients corresponding to multiple frequencies are no longer minors of the determinant, which, as is known, turn into in this case to zero.
Each multiple (or, as they say, degenerate) frequency corresponds to as many different normal coordinates as the degree of multiplicity, but the choice of these normal coordinates is not unambiguous. Since the normal coordinates (with the same ) enter the kinetic and potential energies in the form of identically transformable sums, they can be subjected to any linear transformation that leaves the sum of squares invariant.
It is very simple to find normal coordinates for three-dimensional vibrations of one material point located in a constant external field. By placing the origin of the Cartesian coordinate system at the point of minimum potential energy, we obtain the latter in the form of a quadratic form of the variables x, y, z, and the kinetic energy
(m is the mass of particles) does not depend on the choice of direction of the coordinate axes. Therefore, by appropriate rotation of the axes, it is only necessary to bring the potential energy to a diagonal form. Then
and vibrations along the x, y, z axes are the main ones with frequencies
In the special case of a centrally symmetric field, these three frequencies coincide (see Problem 3).
The use of normal coordinates makes it possible to reduce the problem of forced oscillations of a system with several degrees of freedom to problems of one-dimensional forced oscillations. The Lagrange function of the system, taking into account the variable external forces acting on it, has the form
(23,15)
where is the Lagrangian function of free oscillations.
By introducing normal coordinates instead of coordinates, we get:
where the designation is introduced
Accordingly, the equations of motion
(23.17)
Tasks
1. Determine the oscillations of a system with two degrees of freedom if its Lagrange function
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