Linear oscillatory systems with two degrees of freedom. Small free oscillations of a system with two degrees of freedom. Go to main coordinates
Let a system with two degrees of freedom be given and are generalized coordinates. Kinetic and potential energy system is given by formulas (10.2):
The functions T and P are definitely positive, and therefore:
Substituting (10.2) into (10.12), we get differential equations small oscillations of a system with two degrees of freedom:
The system has a zero solution A=B=0, corresponding sustainable situation balance. For non-zero solutions, we compose from (10.15) the relation:
Quadratic (with respect to ) equation (10.18), due to stability inequalities, has two real positive roots. Let's arrange them in ascending order:
For the second main vibration:
| (10.21) |
The main vibrations are harmonic vibrations.
Substituting and in turn in (10.16), we find the connections between amplitudes A and B in the main vibrations: . The factors are called eigenform coefficients (amplitude distribution coefficients). They can be both positive and negative. When both coordinates in the main oscillation are in the same phase; at - in antiphase.
The resulting movement along each coordinate will be the sum of two main oscillations:
| (10.22) |
where - depend on the initial conditions, - do not depend on the initial conditions and are determined by the parameters of the oscillatory system itself. In the general case, the frequencies and are incommensurable, and therefore the resulting motion will not be periodic.
1. Determine the natural frequencies and natural modes of vibration (small) of a double mathematical pendulum formed by two material points equal mass m and two rods each long.
A similar system in general view was considered in Example 2 (§34). Let us use formulas (2) and (3) obtained there.
When , we get:
Since the oscillations are small, then, up to small ones of the second order inclusive:
| (3) |
Taking into account (3) from (1), we note:
| (4) |
Comparing (4) and (2), we notice:
Expanding equation (7.52) of frequencies, we obtain:
From (9.50) we find the distribution coefficients: .
First major oscillation:
Movement in phase - at every moment the rods rotate in one direction.
Second main hesitation:
Movement in antiphase - at every moment the rods rotate in exactly opposite directions.
The vibration modes are shown in Fig. 50. In the second main vibration there is a special point F, which remains motionless. Such points are called nodes. The end point O is not a node.
2. Two solids with masses and and two springs, stiffness and , are combined into a system that is located on a smooth horizontal plane and can perform small linear oscillations.
First major oscillation:
Bodies move in phase, either to the right or to the left. The oscillation amplitude of the second body is 1.62 times greater.
Second main hesitation:
The bodies move in antiphase: either towards each other, towards the node, or diverge from the node. The amplitude of oscillations of the second body is 0.62 of the amplitude of the first.
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 free vibrations 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.
The Rayleigh 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. With monoharmonic (one-tone) free oscillations of an elastic system with a 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 oscillation shape in question (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)
(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 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 for bending 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
THEORETICAL MECHANICS
UDC 531.8:621.8
D.M. Kobylyansky, V.F. Gorbunov, V.A. Gogolin
COMPATIBILITY OF ROTATION AND VIBRATIONS OF BODIES WITH ONE DEGREE OF FREEDOM
Consider a flat body T, on which three perfect connections, preventing only body movements in all directions, as shown in Fig. 1a. The connections are points A, B, C, located at the vertices of an equilateral triangle. Having chosen a coordinate system so that its center coincides with the center of the triangle and is aligned with it (Fig. 1a), we have the coordinates of the connections: A(0;R), B(^l/3 /2; -R/2), C ^-Ld/e /2; -I/2), where I is the distance from the center of the triangle to its vertices, that is, the radius of the circle passing through points A, B, C. In this position, the body will have one degree of freedom only if the normals to its boundary at points A, B, C intersect at one point, which will be the instantaneous center of velocities. Otherwise, the number of degrees of freedom of the body is zero and it cannot not only move translationally, but also perform rotational motion. When a body has one degree of freedom, it can begin to rotate with the instantaneous center of rotation at the intersection point of the above normals. Let this point be the origin of coordinates, point O. If the instantaneous center of rotation does not change its position, then the only possible shape of the body T is a circle of radius R with the center at point O.
The problem arises: are there other forms of the body that allow it to rotate relative to some moving center so that the
did the body of the body continuously pass through three points A, B, C without breaking these connections? In the literature known to us, such a problem has not been considered and, apparently, is being solved for the first time.
To solve this problem, we first consider the movement of triangle ABC as a rigid body, relative to the X1O1Y1 coordinate system associated with the body T (Fig. 1b). Then, if the movement of the triangle occurs in such a way that its vertices continuously remain on the boundary of the body during a complete rotation of the triangle by 360°, then the body will also perform the required movement in reverse relative to the fixed triangle ABC and the associated coordinate system XOU.
We define the movement of the triangle ABC as a rotation relative to the center O and a movement of the center O along the ОіХі axis by /(g), along the ОіУі axis by g(t). Then the parametric equation of the trajectory of point A will have the form: x = ryaSh +/(r); уі=г-єо,?ґ +g(t), ґє (1)
Since at g=0 point O must coincide with point O1, then the condition /(0)= g(0)=0 must be satisfied. We require that when rotated through an angle r = 2n/3, point A coincides with point B1, point B coincides with point C, and point C
With point A1. When turning through an angle r = 4n/3, point A should go to point C1, point B to point A1, and point C to point B1. Combining these requirements for the movement of the vertices of the triangle leads to conditions for the values of the functions of moving the center of rotation /(0)=/(2 p/3)=/(4 p/3)=0; g0)=g(2l/3)=g(4l/3)=0 . (2) Conditions (2) are satisfied by a wide class of functions, in particular functions of the form sin(3mt/2), where m is an integer, and their linear combinations with generally variable coefficients of the form:
H (g) = ^ bt (g) 8Іп(3тґ / 2)
Moreover, as
Fig.1. Calculation scheme: a) - position of the stationary body and its connections in the XOU system; b) - the position of the fixed system X1O1U1 associated with the body, and the movable system XOU associated with the triangle ABC
Fig.2. Shapes of bodies and trajectories of movement of their centers of rotation
Rice. 3. The position of the body when turning at an angle and the corresponding trajectory of movement of its center of rotation
displacement functions, functions that define closed curves, such as cycloids, trochoids, lemniscates, with parameters suitable according to condition (2) can be taken. At the same time, everything possible functions must be periodic with a period of 2p/3.
Thus, the system of parametric equations (1) with conditions on the values of the functions /(^, g(t) (2) or in their form (3) gives the desired equation for the boundary of the body T. Figure 2 shows examples of possible body shapes that satisfy conditions of the task. In the center of each figure the trajectory of the center of rotation O1 is shown, and the point connections A, B, C are enlarged for their better visualization. These examples show that even simple types of functions from the class defined by expression (3) with constant coefficients give. we have a fairly wide set of curves describing the boundaries of bodies undergoing rotation and
oscillations simultaneously with only one degree of freedom. Boundary curves a), c) in Fig. 2 correspond to the movement of the center of rotation only along the horizontal axis
ОіХі according to the harmonic law, and as can be seen, have two axes of symmetry and can be either purely convex, oval (Fig. 2a), or combine convexity with concavity (Fig. 2b). With a vertical and horizontal harmonic law with the same amplitude of movement of the center of rotation, the boundary curves lose their symmetry (Fig. 2 c, d). Significant influence of frequency harmonic vibrations on the shape of the boundary curve of the body is shown in Fig. 2 d, f. Without conducting a full analysis of the influence of amplitude and frequency on the shape and geometric properties of the boundary curves in this work, I would like to note that the examples presented in Fig. 2 already show the possibility of solving technical problems in choice the desired shape
body to combine it rotational movement with oscillations in the plane of rotation.
Considering now the movement of the body relative to the fixed coordinate system XOU associated with the triangle ABC, that is, moving from the X1O1U1 coordinate system to the XOU coordinate system, we obtain the following parametric equations boundary curve of the body at a given angle of rotation p x=cosp-
Cosp(4)
or taking into account equations (1), equations (4) take the form x = cosp-
- [ R cos(t) + g (t) - g (p)] sin p, y = sin p +
Cos p.
Equations (5) make it possible to describe the trajectory of any point of the body according to its given polarities.
t-g.i m*4<. п-і
t-ÍLÍtWM. d-0
Rice. 4. Variants of body shapes with different numbers of connections, ensuring the compatibility of rotation and vibration of bodies
nal coordinates R,t. In particular, at R=0, t=0 we have a point coinciding with the origin of coordinates Ob, that is, the center of rotation, the trajectory of which in the scheme under consideration is described by the equations following from (5):
*0 = -f (ph) cos ph + g (ph) sin ph, y0 = - f (ph) sin ph- g (ph) cos r.
Figure 3 shows an example of body positions (Figure 2b) when it is rotated through an angle φ, and in the center of each figure the trajectory of the center of rotation is shown
Oi, corresponding to the rotation of the body through this angle. Technically it is not difficult to make animation
of the body movement shown in Fig. 3 instead of a physical model, however, the framework of a journal article can only allow this in an electronic version. The example shown was still
A generalization of the problem considered is a system of n ideal connections in the form of points located at the vertices of a regular triangle, preventing only translational movements of the body. Therefore, as in the case of a triangle, the body can begin to rotate relative to the center of rotation, which is the point of intersection of the normals to the boundary of the body at the connection points. In this case, the equation for the trajectory of a point of body A, located on the axis OU, and located at a distance H from the center of rotation, will have the same form as (1). The conditions for the values of the functions of moving the center of rotation (2) in this case will take
Kobylyansky Gorbunov
Dmitry Mikhailovich Valery Fedorovich
Postgraduate student of the department. stationary and - doc. tech. sciences, prof. department hundred
transport vehicles, stationary and transport vehicles
f(2kp/p)=g(2kp/p)=0. (7)
Condition (7) corresponds to periodic functions with a period of 2n/n, for example 8m(n-m4/2), as well as their linear combinations of the form (3) and other functions describing closed curves. Reasoning similar to that mentioned above leads to the same equations (4-6), which make it possible to calculate the shape of the body, its position during rotation and the trajectory of the center of rotation with oscillations of the body consistent with the rotation. An example of such calculations is Fig. 4, in which the dotted line shows the initial position of the bodies, the solid line shows the position of the bodies when rotating through an angle l/3, and in the center of each figure is the complete trajectory of the center of rotation during a full rotation of the body. And although in this example only the horizontal movement of the center of rotation O, as the center of a n-gon, is considered, the results obtained show a wide range of possible shapes of a body with one degree of freedom, combining rotational motion with oscillations in the presence of four, five and six connections.
The resulting method for calculating the compatibility of rotation and oscillation movements of bodies with one degree of freedom can also be used without any additions for spatial bodies for which movements along the third coordinate and rotations in other coordinate planes are prohibited.
Gogolin Vyacheslav Anatolievich
Dr. tech. sciences, prof. department applied mathematician and
Let us consider small oscillations of a system with two degrees of freedom, which is subject to the forces of a potential field and forces that periodically change in time. The resulting movements of the system are called forced oscillations.
Let the disturbing generalized forces vary according to a harmonic law with time, having equal periods and initial phase. Then the equations of motion of the system under consideration will be of the form:
The equations of motion in the case under consideration are a system of linear second-order differential equations with constant coefficients and a right-hand side.
Go to main coordinates
For the convenience of studying the equations of motion, let us move on to the main coordinates of the system. The relationship between the coordinates is determined by the formulas of the previous paragraph of the form:
Let us denote by correspondingly the generalized forces corresponding to the normal coordinates. Since the generalized forces represent coefficients for the corresponding variations of the generalized coordinates in the expression of the elementary work of forces acting on the system, then
Hence:
Thus, the equations of motion in principal coordinates take the form:
The equations of forced oscillations of a system with two degrees of freedom in normal coordinates are independent of each other and can be integrated separately.
Critical frequencies of the disturbing force
The equation for or determines the oscillatory nature of the change in normal coordinates, studied in detail when considering the forced oscillation of a point along a straight line, since the differential equations of motion are the same in both cases. In particular, if the frequency of the disturbing force is equal to the frequency of one of the natural oscillations of the system, or then the solution will include time t as a factor. Consequently, one of the normal generalized coordinates for a sufficiently large t will be arbitrarily large, or we have the phenomenon of resonance.
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. Hence, every free rigid body 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 refers to the most mobile joints of the human body, which have 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, it is influenced by a number of factors related to age, gender, individual characteristics, the functional state of the nervous system, the degree of muscle stretching, ambient temperature, 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 the 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, regarding 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.