Oscillations with several degrees of freedom.
Brief information from the theory.
Systems with n powersfreedom in dynamics it is customary to call such systems, in order to completely fix the geometric state of which at any moment in time it is necessary to set P parameters, for example position (deflections) P points. The position of other points is determined by conventional static techniques.
An example system with P degrees of freedom can be a beam or a flat frame if the masses of its individual parts or elements are conditionally (to facilitate dynamic calculations) considered concentrated in P points, or if it carries n large masses (engines, motors), in comparison with which it is possible to neglect the own weight of the elements. If individual concentrated (“point”) masses can, when oscillating, move in two directions, then the number of degrees of freedom of the system will be equal to the number of connections that should be imposed on the system in order to eliminate the displacements of all masses.
If a system with n degrees of freedom is brought out of equilibrium, it will commit free vibrations, and each “point” (mass) will perform complex polyharmonic oscillations of the type:
Constants A i and B i depend on the initial conditions of motion (deviations of masses from the static level and speeds at the moment of time t=0). Only in some special cases of excitation of oscillations can polyharmonic motion for individual masses turn into harmonic, i.e. as in a system with one degree of freedom:
The number of natural frequencies of a system is equal to the number of its degrees of freedom.
To calculate natural frequencies, it is necessary to solve the so-called frequency determinant, written in this form:
This condition in expanded form gives the equation P th degree to determine P values of ω 2, which is called the frequency equation.
Through δ 11, δ 12, δ 22, etc. possible movements are indicated. Thus, δ 12 is the displacement in the first direction of the point of location of the first mass from a unit force applied in the second direction to the point of location of the second mass, etc.
With two degrees of freedom, the frequency equation takes the form:
Where for two frequencies we have:
In the case when individual masses M i can also perform rotational or only rotational movements in combination with linear movements, then i-that coordinate will be the angle of rotation, and in the frequency determinant the mass
M i must be replaced by the moment of inertia of the mass J i; accordingly, possible movements in the direction i-th coordinates ( δ i 2 , δ i 2 etc.) will be angular movements.
If any mass oscillates in several directions - i-mu and k-th (for example, vertical and horizontal), then such a mass participates in the determinant several times under the numbers M i them k and it corresponds to several possible movements ( δ ii, δ kk, δ ik, etc.).
Note that each natural frequency has its own special form of oscillation (the nature of a curved axis, line of deflection, displacement, etc.), which in individual, special cases may turn out to be a valid form of oscillation, if only free oscillations are properly excited (proper selection impulses, points of their application, etc.). In this case, the system will oscillate according to the laws of motion of the system with one degree of freedom.
In the general case, as follows from expression (9.1), the system performs polyharmonic oscillations, but it is obvious that any complex elastic line, which reflects the influence of all natural frequencies, can be decomposed into individual components of the form, each of which corresponds to its own frequency The process of such decomposition of the true mode of vibration into components (which is necessary when solving complex problems of structural dynamics) is called decomposition into the modes of natural vibrations.
If in each mass, more precisely - in the direction of each degree of freedom, a disturbing force is applied, varying in time according to the harmonic law
or, which is indifferent for further purposes, and the amplitudes of the forces for each mass are different, and the frequency and phases are the same, then with prolonged action of such disturbing forces the system will perform steady-state forced oscillations with the frequency of the driving force. Amplitudes of movements in any direction i-that degree in this case will be:where the determinant D is written according to (9.2) with ω replaced by θ and, therefore, D≠0; D i is determined by the expression:
those. i The th column of the determinant D is replaced by a column composed of terms of the form: For the case of two degrees of freedom: (9.6)And correspondingly
When calculating forced vibrations of beams of constant cross-section carrying concentrated masses (Fig. 9.1).
It is easier, however, to use the following formulas for the amplitudes of deflection, angle of rotation, bending moment and shear force in any section of the beam:
(9.7)Where y 0 , φ 0 , M 0 , Q 0 – amplitudes of deflection, rotation, moment and shear force of the initial section (initial parameters); M i And J i- mass and its moment of inertia (concentrated masses); the sign ∑ applies to all forces and concentrated masses located from the initial section to the subject.
The indicated formulas (9.7) can also be used when calculating natural frequencies, for which it is necessary to consider the disturbing forces ∑ Ri and moments ∑ Mi equal to zero, replace the frequency of forced oscillations θ with the frequency of natural oscillations ω and, assuming the existence of oscillations (free oscillations), write expressions (9.7) in relation to sections where concentrated masses are located and amplitudes are already known (reference sections, axis of symmetry, etc. ). We obtain a system of homogeneous linear equations. Equating the determinant of this system to zero, we will be able to calculate the natural frequencies.
It turns out to be advisable to use expressions (9.4) and (9.5) to determine the amplitudes ( y 0 , φ 0 , etc.) at X=0, and then using (9.7) calculate all other deflection elements.
More complex is the problem of calculating the motions of a system with several degrees of freedom under the action of an arbitrary load that changes over time and is applied to various masses.
When solving such a problem, you should proceed as follows:
a) determine natural frequencies and modes of natural vibrations;
b) rearrange the given load between masses or, as they say, decompose it into its own modes of vibration. The number of load groups is equal to the number of natural frequencies of the system;
c) after performing the above two auxiliary operations, make a calculation for each group of loads using known formulas from the theory of oscillations of a system with one degree of freedom, and the frequency of natural oscillations in these formulas is taken to be the one to which this load group corresponds;
d) partial solutions from each category of loads are summed up, which determines the final solution of the problem.
Determination of natural frequencies is carried out according to (9.2). As for identifying the forms of natural vibrations, here it is necessary to be guided by the basic property of any form of natural vibrations, that it represents the line of influence of the deflection from forces (the number of which is equal to the number of degrees of freedom) proportional to the product of the masses and the ordinates of the deflections of the points of attachment of the masses. For equal masses, the form of natural vibrations represents the line of deflection from forces proportional to the ordinates of the deflection; the load diagram is similar to the deflection diagram.
The lowest frequency corresponds to the simplest form of vibration. For beams, most often this shape closely corresponds to the curved axis of the system under the influence of its own weight. If this structure turns out to be less rigid in any direction, for example in the horizontal, then to identify the nature of the desired curved axis, one must conditionally apply its own weight in this direction.
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.
According to the general rules for solving 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 that 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 factor 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)
1. Determine the oscillations of a system with two degrees of freedom if its Lagrange function
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 obtain a linear system of algebraic equations for A 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 system equations 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. Case of multiple frequencies k ( = k. ) , as well as the turning of the 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 (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 particular solutions of the form (4.23), so the 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 system parameters 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 form factor value means that the oscillations are out of phase.
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.
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
Let us consider a flat body T, on which three ideal constraints are imposed, preventing only the movement of the body 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)
In addition, 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
Theoretical mechanics
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. In this case, all possible functions must be periodic with a period of 2n/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 performing 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). The significant influence of the frequency of harmonic vibrations on the shape of the boundary curve of a 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 ability to solve technical problems in choosing the desired form
body to combine its rotational motion 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 of the 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
