Foundations of the theory of vibrations of mechanical systems. Fundamentals of vibration theory

Ministry of Education of the Russian Federation
Ukhta State Technical University

VC. Khegai, D.N. Levitsky,
HE. Kharin, A.S. Popov

Fundamentals of vibration theory
mechanical systems
Tutorial

Approved by the educational and methodological association of universities
in higher oil and gas education as an educational
manuals for students of oil and gas universities studying
in the specialty 090800, 170200, 553600

UDC 534.01
X-35
Fundamentals of the theory of oscillations of mechanical systems / V.K. Hegai,
D.N. Levitsky, O. N. Kharin, A.S. Popov. - Ukhta: USTU, 2002 .-- 108 p.
ISBN 5-88179-285-8
The tutorial discusses the basics of the theory of oscillations of mechanical systems, which are based on general course theoretical mechanics... Particular attention is paid to the application of the Lagrange equations of the second
row. The manual consists of six chapters, each of which is devoted to a specific type of vibration. One chapter is devoted to the foundations of the theory of stability of motion and equilibrium of mechanical systems.
For better mastering theoretical material, in the manual, given
a large number of examples and problems from various fields of technology.
The textbook is intended for students of mechanical specialties studying the course of theoretical mechanics in full,
it can also be useful for students of other specialties.
Reviewers: Department of Theoretical Mechanics, St. Petersburg
State Forestry Academy (Head of the Department, Doctor of Technical Sciences, Professor YA Dobrynin); Head of the Integrated Drilling Department of SeverNIPIGaz, Candidate of Technical Sciences, Associate Professor Yu.M. Gerzhberg.

3
Table of contents
Foreword ................................................. .................................................. .................. 4
Chapter I. Brief information from analytical mechanics ........................................ 5
1.1 Potential energy of the system .............................................. ................................. 5
1.2. Kinetic energy of the system ............................................... ................................. 6
1.3. Dissipative function ................................................ ............................................ eight
1.4. Langrange equation ................................................ ................................................ nine
1.5. Examples for the preparation of Langrange equations of the second kind ............................. 11
Chapter II. Stability of movement and balance of conservative systems ......... 20
2.1. Introduction ................................................. .................................................. ................... twenty
2.2. Lyapunov functions. Sylvester criterion ................................................ ............. 21
2.3. The equation of disturbed motion ............................................... ......................... 23
2.4. Lyapunov's theorem on the stability of motion ............................................. .......... 26
2.5. Lagrange's theorem on the stability of an equilibrium
conservative system ................................................ .................................................. 29
2.6. The stability of the equilibrium of a conservative system with one
degree of freedom ................................................ .................................................. ........... thirty
2.7. Examples of the stability of the equilibrium of a conservative system ......................... 31
Chapter III. Free vibrations of a system with one degree of freedom ................. 39
3.1. Free vibrations of a conservative system
with one degree of freedom .............................................. ................................................. 39
3.2. Free vibrations of a system with one degree of freedom in the presence
resistance forces proportional to the speed ............................................. ............ 42
3.3. Examples of free vibrations of a system with one degree of freedom ............. 46
Chapter IV. Forced oscillations of a system with one degree of freedom ........... 59
4.1. Forced oscillations of a system with one degree of freedom
in the case of a periodic disturbing force ............................................. ................... 59
4.2. The phenomenon of resonance ................................................ .................................................. .... 63
4.3. Phenomenon of beating ................................................ .................................................. ........ 66
4.4. Dynamic factor ................................................ ..................................... 68
4.5. Examples of forced vibrations of the system
with one degree of freedom .............................................. ................................................. 70
Chapter V. Free oscillations of a system with two degrees of freedom ................ 78
5.1. Differential equations of free oscillations of a system with two
degrees of freedom and their general solution ............................................ ............................ 78
5.2. Own forms ................................................ .................................................. 80
5.3. Examples for free vibration of a system with two degrees of freedom ............ 81
Chapter VI. Forced oscillations of a system with two degrees of freedom ........ 93
6.1. Differential equations of forced oscillations of the system and their
common decision................................................ .................................................. ................. 93
6.2. Dynamic vibration damper ............................................... ........................... 95
6.3. Examples of forced vibrations of a system with two degrees of freedom ... 98
Bibliographic list ................................................ .......................................... 107

4
Foreword
At the present stage of development high school problem and research forms of education are being introduced into teaching practice.
Dynamic processes in machines and mechanisms are of decisive importance both for the calculation at the design stage of new structures and for the determination of technological modes during operation. It is difficult to name a field of technology in which there would not be
topical problems of studying elastic vibrations and stability of equilibrium and motion of mechanical systems. They represent a special
importance for mechanical engineers working in mechanical engineering, transport and other fields of technology.
The manual examines some of the specific issues from the theory
vibrations and stability of mechanical systems. Theoretical information
are explained by examples.
The main purpose of this methodological manual- to link
application area of theoretical and analytical mechanics with problems
special departments that train mechanical engineers.

5
Chapter I. BRIEF INFORMATION FROM ANALYTICAL
MECHANICS
I.I. Potential energy of the system
The potential energy of a system with s degrees of freedom, being
position energy depends only on the generalized coordinates

P = P (q1, q2, ....., qs),
where q j

(j = 1, 2, K, s) - generalized coordinates of the system.

Considering small deviations of the system from the position of a stable
equilibrium, generalized coordinates qj can be considered as quantities of the first order of smallness. Assuming that the equilibrium position of the system
corresponds to the origin of the generalized coordinates, we expand the expression for the potential energy P in the Maclaurin series in powers of qj

∂П
1 S S ∂2 П
P = P (Ο) + ∑ (
) 0 q j + ∑∑ (
) 0 qi q j + K.
∂
q
2
∂
q
∂
q
j = 1
i = 1 j = 1
j
i
j
S

Bearing in mind that the potential energy is determined with an accuracy
to some additive constant, the potential energy in the equilibrium position can be taken equal to zero
P (0) = 0.

In the case of conservative forces, the generalized forces are determined by the formula

∂П
∂q j

(j = 1, 2, K, s).

Since at equilibrium of the system of forces

(j = 1, 2, K, s),

Then the conditions of equilibrium of the conservative system of forces have the form

⎛ ∂П
⎜⎜
⎝ ∂q j

⎞
⎟⎟ = 0
⎠0

(j = 1, 2, K, s),

⎛ ∂П
∑⎜
j = 1 ⎜ ∂q
⎝ j

⎞
⎟⎟ q j = 0.
⎠0

Hence,
s

6
Then equality (1.2.), Up to terms of the second order of smallness, takes the form

1 S S ⎛ ∂2 П
P = ∑∑⎜
2 i = 1 j = 1 ⎜⎝ ∂qi ∂q j

⎞
⎟⎟ qi q j.
⎠0

We denote

⎛ ∂2 П
⎜⎜
⎝ ∂qi ∂q j

⎞
⎟⎟ = cij = c ji,
⎠0

Where cij are generalized stiffness factors.
The final expression for the potential energy has the form

1 S S
П = ∑∑cij qi q j.
2 i = 1 j = 1

It can be seen from (1.9.) That the potential energy of the system is homogeneous quadratic function generalized coordinates.
1.2. Kinetic energy of the system
The kinetic energy of a system consisting of n material points,
is equal to

1 n
T = ∑mk vk2,
2 k = 1

Where mk and vk are the mass and speed of the k-th point of the system.
When passing to generalized coordinates, we will keep in mind that
_

(k = 1, 2, ..., n),

R k (q1, q2, ..., qs)

Where r k is the radius vector of the k-th point of the system.
−

We use the identity vk2 = v k ⋅ v k and replace the velocity vector
−

V k by its value
_

∂r k
∂q1

∂r k
∂q2

∂r k
∂qs

Then the expression for the kinetic energy (1.10) takes the form

7
2
2
2

1
T = (A11 q1 + A22 q 2 + ... + ASS q S + 2 A12 q1 q 2 + ... + 2 AS −1, S q S −1 q S), (1.13)
2

⎛ _
∂ rk
A11 = ∑ mk ⎜
⎜ ∂q1
k = 1
⎝
n

⎛ _
∂ rk
Ass = ∑ mk ⎜
⎜ ∂qs
k = 1
⎝
n

⎞
⎛ _
n
⎟, A22 = ∑ mk ⎜ ∂ r k
⎟
⎜ ∂q2
k = 1
⎠
⎝

⎞
⎟ ,...,
⎟
⎠

_
_
⎞
r
r
∂
∂
⎟, A12 = ∑ mk k ⋅ k, ...,
⎟
∂q1 ∂q2
⎠
_

As −1, s = ∑ mk
k = 1

∂ rk ∂ rk
.
⋅
∂qS −1 ∂qS

Expanding each of these coefficients in a Maclaurin series in powers of the generalized coordinates, we obtain

⎛ ∂Aij
Aij = (Aij) 0 + ∑ ⎜
⎜
j = 1 ⎝ ∂A j
S

⎞
⎟⎟ q j + ...
⎠0

(i = j = 1, 2, ..., s).

Index 0 corresponds to the values of the functions in the equilibrium position. Since we are considering small deviations of the system from the position
equilibrium, then in equality (1.14) we restrict ourselves only to the first constant terms

(i = j = 1, 2, ..., s).

Aij = (Aij) 0 = aij

Then the expression for the kinetic energy (1.13) takes the form
2
2

⎞
1⎛ 2
T = ⎜ a11 q1 + a22 q 2 + ... + aSS q S + 2a12 q1 q 2 + 2aS −1, S q S −1 q S ⎟ (1.15)
2⎝
⎠

Or in general terms

1 S
T = ∑
2 i = 1

Constants aij - generalized coefficients of inertia.
It is seen from (1.16) that the kinetic energy of the system T is uniform
quadratic function of generalized velocities.

8
1.3. Dissipative function
In real conditions, the free oscillations of the system are damped, so
how the forces of resistance act on its points. In the presence of resistance forces, mechanical energy dissipates.
−

Let us assume that the resistance forces R k (k = 1, 2, ..., n) acting
to the points of the system, proportional to their speeds
_

R k = - μk v k

(k = 1, 2, ..., n),

Where µ k is the proportionality coefficient.
The generalized drag forces for the holonomic system are determined by the formulas
n

Q j R = ∑ Rk
k = 1

∂ rk
∂r
= −∑ µ k vk k
∂q j
∂q j
k = 1
n

(j = 1, 2, ..., s).

Because
_

∂ rk
∂ rk
∂ rk
q1 +
q 2 + ... +
qS,
∂q1
∂q2
∂qS

∂ rk
.
∂q j

Bearing in mind (1.18), we rewrite the generalized resistance forces (1.17) in the form
n

Q = −∑ µκ vκ
R
j

(j = 1, 2, ..., s).

Let us introduce a dissipative function, which is determined by the formula
n

Then the generalized resistance forces are determined by the formulas

(j = 1, 2, ..., s).

By analogy with the kinetic energy of the system, the dissipative function can be represented as a homogeneous quadratic function
generalized speeds

1 S S
Φ = ∑∑ вij q i q j,
2 i = 1 j = 1

Where вij are the generalized dissipation coefficients.
1.4. Lagrange equation of the second kind
The position of a holonomic system with s degrees of freedom is determined by s generalized coordinates qj (j = 1, 2, ..., s).
To derive the Lagrange equations of the second kind, we use the general
dynamic equation
S

Q and j) δ q j = 0,

Where Qj is the generalized force of active forces corresponding to the j-th generalized coordinate;
Q uj - generalized force of inertia forces corresponding to the j-th generalized coordinate;
δ q j - increment of the j -th generalized coordinate.
Bearing in mind that all δ q j (j = 1, 2, ..., s) are independent of each other,
equality (1.23) will be valid only in the case when each of the coefficients at δ q j separately equals zero, i.e.

Q j + Q and j = 0 (j = 1, 2, ..., s)
or

(j = 1, 2, ..., s).

Let us express Q uj in terms of the kinetic energy of the system.
By the definition of the generalized force, we have

Q and j = ∑ Φ k
k = 1

∂ rk
d vk ∂ r k
= - ∑ mk
⋅
1
=
k
∂q j
dt ∂q j
n

(j = 1, 2, K, s),

D vk
where Φ k = - mk a k = - mk
Is the force of inertia to the th point of the system.
dt
_

⎛_ _
d vk ∂ r k d ⎜ ∂ r k
⋅
=
vk ⋅
⎜
dt ∂q j dt
∂q j
⎝
_

⎞ _
⎛ _
⎟ - vk ⋅ d ⎜ ∂ r k
⎟
dt ⎜ ∂q j
⎠
⎝

⎞
⎟,
⎟
⎠

R k = r k (q1, q2, ..., qs),
_

D rk ∂ rk
∂ rk
∂ rk
vk =
=
q1 +
q 2 + ... +
qs,
dt
∂q1
∂q2
∂q s
_

⎛ _
d ⎜ ∂ rk
dt ⎜ ∂q j
⎝

_
_
⎞
∂
d
r
∂
v
k
k
⎟=
=
.
⎟ ∂q j dt
∂q j
⎠

Substituting the values (1.27) and (1.28) into equality (1.26), we find
_
⎛_
∂ vk ∂ r k d ⎜
∂ vk
vk ⋅
⋅
=
∂t ∂q j dt ⎜⎜
∂qj
⎝
_

_
⎞ _
⎛
∂ vk2
∂
v
d
k
⎜
⎟ - vk ⋅
=
⎟⎟
∂q j dt ⎜⎜ 2∂ q
j
⎠
⎝

⎞
2
⎟ - ∂ vk.
⎟⎟ 2∂q j
⎠

Taking into account equality (1.29), expression (1.25) can be rewritten as

⎡ ⎛
d ⎜ ∂vk2
and
⎢
−Q j = ∑ mk
⎢ dt ⎜⎜
k = 1
⎣⎢ ⎝ 2∂ q j
n

∂
−
∂q j

⎞
⎛
2 ⎤
v
∂
d⎜ ∂
k ⎥
⎟−
=
⎥
⎟⎟
dt ⎜⎜ ∂ q
2
q
∂
j ⎦
j
⎥
⎠
⎝

⎛
mk vk2 d ⎜ ∂Τ
=
∑
2
dt ⎜⎜ ∂ q
k = 1
j
⎝
n

⎞
⎟ − ∂Τ .
⎟⎟ ∂q j
⎠

⎞
mk vk2 ⎟
−
∑
2 ⎟⎟
k = 1
⎠
n

11
Here it is taken into account that the sum of derivatives is equal to the derivative of the sum,
n m v2
and ∑ k k = T is the kinetic energy of the system.
k = 1
2
Bearing in mind equalities (1.24), we finally find

⎛
d ⎜ ∂Τ
dt ⎜⎜ ∂ q
⎝ j

⎞
⎟ - ∂Τ = Q
j
⎟⎟ ∂q j
⎠

(j = 1, 2, K, s).

Equations (1.30) are called Lagrange equations of the second kind.
The number of these equations is equal to the number of degrees of freedom.
If the forces acting on the points of the system have a potential, then
for generalized forces, the following formula is valid

∂П
∂q j

(j = 1, 2, K, s),

Where P is the potential energy of the system.
Thus, for the conservative system of the Lagrange equation

The book introduces the reader to general properties oscillatory processes occurring in radio engineering, optical and other systems, as well as with various qualitative and quantitative methods of studying them. Considerable attention is paid to the consideration of parametric, self-oscillating and other nonlinear oscillatory systems.
The study of the oscillatory systems and processes in them described in the book is given by the well-known methods of the theory of oscillations without a detailed presentation and justification of the methods themselves. The main attention is paid to clarifying the fundamental features of the studied oscillatory models of real systems using the most adequate methods of analysis.

Free oscillations in a circuit with non-linear inductance.
Consider now another example of an electrical nonlinear conservative system, namely, a circuit with an inductance that depends on the current flowing through it. This case does not have a clear and simple nonrelativistic mechanical analogue, since the dependence of self-induction on the current is equivalent for mechanics to the case of the dependence of mass on velocity.

We meet electrical systems of this type when cores made of ferromagnetic material are used in inductors. In such cases, for each given core, you can get the relationship between the magnetizing zero and the flux of magnetic induction. The curve depicting this relationship is called the magnetization curve. If we neglect the phenomenon of hysteresis, then its approximate course can be represented by the graph shown in Fig. 1.13. Since the magnitude of the field H is proportional to the current flowing in the coil, the current can be plotted along the abscissa axis directly on the appropriate scale.

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Download the book Fundamentals of the theory of vibrations, Migulin V.V., Medvedev V.I., Mustel E.R., Parygin V.N., 1978 - fileskachat.com, fast and free download.

Principles of theoretical physics, Mechanics, field theory, elements of quantum mechanics, Medvedev B.V., 2007
Physics course, A.P. Ershov, G.V. Fedotovich, V.G. Kharitonov, E.R. Pruuel, D.A. Medvedev
Technical thermodynamics with the basics of heat transfer and hydraulics, Lashutina N.G., Makashova O.V., Medvedev R.M., 1988

We have already examined the origin of classical mechanics, resistance of materials and the theory of elasticity. The most important component of mechanics is also the theory of oscillations. Vibrations are the main cause of destruction of machinery and structures. By the end of the 1950s. 80% of equipment accidents occurred due to increased vibrations. Vibrations also have a harmful effect on people associated with the operation of equipment. They can also cause control system failures.

Despite all this, the theory of oscillations emerged as an independent science only at the turn of the 19th century. However, the calculations of machines and mechanisms up to the beginning XX centuries were held in a static setting. The development of mechanical engineering, an increase in the power and speed of steam engines with a simultaneous decrease in their weight, the emergence of new types of engines - internal combustion engines and steam turbines led to the need for strength calculations taking into account dynamic loads. As a rule, new problems in the theory of vibrations arose in technology under the influence of accidents or even catastrophes resulting from increased vibrations.

Oscillations are movements or changes in state that have varying degrees of repeatability.

Oscillation theory can be divided into four periods.

Iperiod- the emergence of the theory of oscillations within the framework of theoretical mechanics (end of the 16th century - the end of the 18th century). This period is characterized by the emergence and development of dynamics in the works of Galileo, Huygens, Newton, d "Alambert, Euler, D. Bernoulli and Lagrange.

The founder of the theory of oscillations was Leonard Euler. In 1737, L. Euler, on behalf of the St. Petersburg Academy of Sciences, began research on the balance and motion of a ship, and in 1749 his book "Ship Science" was published in St. Petersburg. It was in this work of Euler that the foundations of the theory of static stability and the theory of oscillations were laid.

Jean Leron d "Alambert, in his numerous works, considered individual problems, such as small oscillations of a body around the center of mass and around the axis of rotation in connection with the problem of the precession and nutation of the Earth, oscillations of a pendulum, a floating body, a spring, etc. But general theory hesitation d "Alambert did not create.

The most important application of the methods of the theory of vibrations was the experimental determination of the torsional stiffness of the wire, carried out by Charles Coulomb. Empirically, Coulomb also established the property of isochronism of small oscillations in this problem as well. Investigating the damping of oscillations, this great experimenter came to the conclusion that its main cause is not air resistance, but losses from internal friction in the wire material.

A great contribution to the foundations of the theory of oscillations was made by L. Euler, who laid the foundations of the theory of static stability and the theory of small oscillations, d 'Alambert, D. Bernoulli and Lagrange. In their works, the concepts of the period and frequency of oscillations, modes of oscillation were formed, the term small oscillations came into use. , the principle of superposition of solutions was formulated, attempts were made to expand the solution into a trigonometric series.

The first problems in the theory of oscillations were the problems of oscillations of a pendulum and a string. We have already spoken about the oscillations of the pendulum - the practical result of solving this problem was the invention of the clock by Huygens.

As for the problem of string vibrations, this is one of the most important problems in the history of the development of mathematics and mechanics. Let's consider it in more detail.

String acoustics it is an ideal straight, thin and flexible thread of finite length made of solid material, stretched between two fixed points. V modern interpretation the problem of transverse vibrations of a string of length l is reduced to finding a solution to the differential equation (1) in partial derivatives. Here x Is the coordinate of the point of the string along the length, and y- its lateral displacement; H- string tension, - its linear mass. a this is the speed of propagation of the wave. A similar equation also describes the longitudinal vibrations of the air column in the pipe.

In this case, the initial distribution of the deviations of the points of the string from a straight line and their velocities should be specified, i.e. equation (1) must satisfy the initial conditions (2) and boundary conditions (3).

The first fundamental experimental studies of string vibrations were carried out by the Dutch mathematician and mechanic Isaac Beckmann (1614–1618) and M. Mersenne, who established a number of regularities and published his results in 1636 in the "Book of Consonances":

Mersenne's laws were theoretically confirmed in 1715 by Newton's student Brook Taylor. He considers the string as a system of material points and accepts the following assumptions: all points of the string simultaneously pass their equilibrium positions (coincide with the axis x) and the force acting on each point is proportional to its displacement y about the axis x... This means that he reduces the problem to a system with one degree of freedom - equation (4). Taylor correctly received the first natural frequency (fundamental) - (5).

D "Alambert in 1747 for this problem applied the method of reducing the problem of dynamics to the problem of statics (principle d" Alambert) and obtained the differential equation of vibrations of a homogeneous string in partial derivatives (1) - the first equation of mathematical physics. He sought the solution of this equation in the form of the sum of two arbitrary functions (6)

where and - periodic functions of period 2 l... When clarifying the question about the type of functions and e "Alambert takes into account the boundary conditions (1.2), assuming that for
the string is aligned with the axis x... The meaning is
not specified in the problem statement.

Euler considers a special case when for
the string is deflected from the equilibrium position and released without initial velocity. It is essential that Euler does not impose any restrictions on the initial shape of the string, i.e. does not require that it can be specified analytically by considering any curve that "can be drawn by hand." The final result obtained by the author: if at
the shape of the string is described by the equation
, then the oscillations look like this (7). Euler revised his views on the concept of a function, in contrast to the previous view of it only as an analytical expression. Thus, the class of functions to be studied in analysis was expanded, and Euler came to the conclusion that "since any function will define a certain line, the opposite is also true - curved lines can be reduced to functions."

The solutions obtained by d'Alembert and Euler represent the law of vibrations of a string in the form of two waves traveling towards each other. However, they did not agree on the form of the function defining the bend line.

D. Bernoulli in the study of string vibrations took a different path, breaking the string into material points, the number of which he considered to be infinite. He introduces the concept of a simple harmonic vibration of the system, i.e. such its movement, in which all points of the system oscillate synchronously with the same frequency, but with different amplitudes. Experiments carried out with sounding bodies led D. Bernoulli to the idea that the most general movement of a string consists in the simultaneous execution of all movements available to it. This is the so-called superposition of solutions. Thus, in 1753, proceeding from physical considerations, he obtained a general solution for string vibrations, presenting it as a sum of particular solutions, for each of which the string bends in the form of a characteristic curve (8).

In this series, the first waveform is half a sinusoid, the second is a whole sinusoid, the third consists of three half-sinusoids, etc. Their amplitudes are represented as functions of time and, in essence, are generalized coordinates of the system under consideration. According to D. Bernoulli's solution, the motion of the string is an infinite series of harmonic vibrations with periods
... In this case, the number of nodes (fixed points) is one less than the number of the natural frequency. Restricting series (8) to a finite number of terms, we obtain a finite number of equations for the continuous system.

However, D. Bernoulli's solution contains an inaccuracy - it does not take into account that the phase shift for each harmonic of oscillations is different.

D. Bernoulli, presenting the solution in the form of a trigonometric series, used the principle of superposition and the expansion of the solution in terms of the complete system of functions. He rightly believed that with the help of various terms in formula (8), one can explain the harmonic tones that the string emits simultaneously with its fundamental tone. He considered this as a general law, valid for any system of bodies performing small vibrations. However, physical motivation cannot replace mathematical proof, which was not presented at the time. Because of this, colleagues did not understand the solution of D. Bernoulli, although as early as 1737 C. A. Clairaut used the expansion of functions in a series.

The presence of two different ways the solution of the problem of vibrations of a string was called among the leading scientists of the XVIII century. stormy controversy - "the dispute about the string." This dispute mainly concerned the questions of what form the admissible solutions of the problem have, the analytical representation of a function, and whether it is possible to represent an arbitrary function in the form of a trigonometric series. In the "dispute about the string" one of the most important concepts analysis - the concept of function.

D "Alambert and Euler did not agree that the solution proposed by D. Bernoulli could be general. In particular, Euler could not agree in any way that this series could represent any" freely drawn curve ", as he himself is now defined the concept of a function.

Joseph Louis Lagrange, having entered into a polemic, broke the string into small arcs of the same length with the mass concentrated in the center, and investigated the solution of a system of ordinary differential equations with a finite number of degrees of freedom. Then passing to the limit, Lagrange obtained a result similar to that of D. Bernoulli, without postulating, however, in advance that the general solution must be an infinite sum of particular solutions. At the same time, he refines the solution of D. Bernoulli, bringing it in the form (9), and also derives formulas for determining the coefficients of this series. Although the decision of the founder of analytical mechanics did not meet all the requirements of mathematical rigor, it was a notable step forward.

As for the expansion of the solution in a trigonometric series, Lagrange believed that the series diverges for arbitrary initial conditions. 40 years later, in 1807, J. Fourier again found the expansion of the function in a trigonometric series for the third time and showed how it can be used to solve the problem, thereby confirming the correctness of D. Bernoulli's solution. A complete analytical proof of the Fourier theorem on the expansion of a single-valued periodic function in a trigonometric series was given in the integral calculus of Todgönther and in "A Treatise on Natural Philosophy" by Thomson (Lord Kelvin) and Theta.

Research into the free vibrations of a stretched string has been going on for two centuries, based on the work of Beckmann. This task served as a powerful stimulus for the development of mathematics. Considering the oscillations of continuous systems, Euler, d "Alambert and D. Bernoulli created a new discipline - mathematical physics. Mathematization of physics, that is, its presentation through a new analysis - Euler's greatest merit, thanks to which new paths were paved in science. Logical development the results of Euler and Fourier was the well-known definition of a function by Lobachevsky and Lejeune Dirichlet, based on the idea of a one-to-one correspondence of two sets. Dirichlet also proved the possibility of Fourier expansion of piecewise continuous and monotone functions. A one-dimensional wave equation was also obtained and the equality of its two solutions was established, which mathematically confirmed the relationship between oscillations and waves. The fact that a vibrating string generates sound prompted scientists to think about the identity of the process of sound propagation and the process of vibrating a string. The most important role of boundary and initial conditions in such problems was also revealed. For the development of mechanics, an important result was the application of the Alambert principle for writing the differential equations of motion, and for the theory of oscillations, this problem also played a very important role, namely, the principle of superposition and the expansion of the solution in terms of natural modes of oscillations were applied, the basic concepts of the theory of oscillations were formulated - natural frequency and mode of vibration.

The results obtained for free vibrations of a string served as the basis for the creation of a theory of vibrations of continuous systems. Further study of the vibrations of inhomogeneous strings, membranes, rods required finding special methods for solving the simplest equations of hyperbolic type of the second and fourth orders.

The problem of free vibrations of a stretched string interested scientists, of course, not for its practical application, the laws of these vibrations were to one degree or another known to the masters of making musical instruments. This is evidenced by the unsurpassed stringed instruments of such masters as Amati, Stradivari, Guarneri and others, whose masterpieces were created in the 17th century. The interests of the greatest scientists involved in this task, most likely, consisted in the desire to bring a mathematical basis for the already existing laws of string vibration. In this issue, the traditional path of any science manifested itself, starting with the creation of a theory that already explains known facts to then find and investigate unknown phenomena.

IIperiod - analytical(late 18th century - late 19th century). The most important step in the development of mechanics was succeeded by Lagrange, who created a new science - analytical mechanics. The beginning of the second period in the development of the theory of oscillations is associated with the work of Lagrange. In his book Analytical Mechanics, published in Paris in 1788, Lagrange summed up everything that was done in mechanics in the 18th century and formulated a new approach to solving its problems. In the theory of equilibrium, he abandoned the geometric methods of statics and proposed the principle of possible displacements (Lagrange's principle). In dynamics, Lagrange, applying simultaneously the principle of d "Alambert and the principle of possible displacements, received a general variational equation of dynamics, which is also called the principle of d" Alambert - Lagrange. Finally, he introduced the concept of generalized coordinates into everyday life and obtained the equations of motion in the most convenient form - the Lagrange equations of the second kind.

These equations became the basis for the creation of the theory of small oscillations described by linear differential equations with constant coefficients. Linearity is rarely inherent in a mechanical system, but in most cases is the result of simplification. Considering small oscillations near the equilibrium position, which are carried out at low speeds, it is possible to discard terms of the second and higher orders relative to the generalized coordinates and velocities in the equations of motion.

Applying the Lagrange equations of the second kind for conservative systems

we get the system s linear differential equations of the second order with constant coefficients

, (11)

where I and C- respectively, the matrices of inertia and stiffness, the components of which will be inertial and elastic coefficients.

A particular solution (11) is sought in the form

and describes a monoharmonic oscillatory mode with a frequency k, which is the same for all generalized coordinates. Differentiating (12) twice with respect to t and substituting the result into equations (11), we obtain a system of linear homogeneous equations for finding the amplitudes in matrix form

. (13)

Since during system oscillations all amplitudes cannot be zero, the determinant is equal to zero

. (14)

Frequency equation (14) is called the secular equation, since it was first considered by Lagrange and Laplace in the theory of secular perturbations of the elements of planetary orbits. It is the equation s degree relative , the number of its roots is equal to the number of degrees of freedom of the system. These roots are usually arranged in ascending order, while they form a spectrum of natural frequencies. To every root corresponds to a particular solution of the form (12), the set s amplitudes represent the waveform, and the overall solution is the sum of these solutions.

Lagrange gave the statement of D. Bernoulli that the general oscillatory motion of a system of discrete points consists in the simultaneous execution of all its harmonic oscillations, the form of a mathematical theorem, using the theory of integration of differential equations with constant coefficients, created by Euler in the 1840s. and the achievements of d "Alambert, who showed how systems of such equations are integrated. In this case, it was necessary to prove that the roots of the secular equation are real, positive and not equal to each other.

Thus, in "Analytical Mechanics" Lagrange obtained the equation of frequencies in general form. At the same time, he repeats the mistake made by d'Alembert in 1761, that the multiple roots of the secular equation correspond to an unstable solution, since in this case secular or secular terms containing t not under the sine or cosine sign. In this regard, both d'Alembert and Lagrange believed that the equation of frequencies cannot have multiple roots (the d'Alembert - Lagrange paradox). It was enough for Lagrange to consider at least a spherical pendulum or vibrations of a rod, the cross-section of which is, for example, round or square, to make sure that multiple frequencies are possible in conservative mechanical systems. The mistake made in the first edition of Analytical Mechanics was repeated in the second edition (1812), published during Lagrange's lifetime, and in the third (1853). The scientific authority of d "Alambert and Lagrange was so high that this mistake was repeated by both Laplace and Poisson, and it was corrected only almost 100 years later, independently of each other, in 1858 by K. Weierstrass and in 1859 by Osip Ivanovich Somov , who made a great contribution to the development of the theory of oscillations of discrete systems.

Thus, to determine the frequencies and forms of free oscillations of a linear system without resistance, it is necessary to solve the secular equation (13). However, equations of degree higher than the fifth do not have an analytical solution.

The problem was not only the solution of the secular equation, but also, in to a greater extent, its compilation, since the expanded determinant (13) has
terms, for example, for a system with 20 degrees of freedom, the number of terms is 2.4 × 10 18, and the time it takes to open such a determinant for the most powerful computer of the 1970s, performing 1 million operations per second, is about 1.5 million years , but for a modern computer "only" a few hundred years.

The problem of determining the frequencies and forms of free vibrations can also be considered as a linear algebra problem and solved numerically. Rewriting equality (13) as

, (14)

note that the column matrix is own matrix vector

, (15)

a by its own meaning.

Solving the problem of eigenvalues and vectors is one of the most attractive problems in numerical analysis. At the same time, it is impossible to propose a single algorithm for solving all problems encountered in practice. The choice of the algorithm depends on the type of matrix, as well as on whether it is necessary to determine all the eigenvalues or only the smallest (largest) or close to a given number. In 1846, Carl Gustav Jacob Jacobi proposed an iterative method of rotations to solve the complete eigenvalue problem. The method is based on such an infinite sequence of elementary rotations, which in the limit transforms matrix (15) into a diagonal one. The diagonal elements of the resulting matrix will be the desired eigenvalues. In this case, to determine the eigenvalues, it is required
arithmetic operations, and for eigenvectors also
operations. In this regard, the method in the XIX century. found no application and was forgotten for more than a hundred years.

The next important step in the development of the theory of vibrations was the work of Rayleigh, especially his fundamental work "Theory of Sound". In this book, Rayleigh examines vibrational phenomena in mechanics, acoustics, and electrical systems from a unified point of view. Rayleigh belongs to a number of fundamental theorems of the linear theory of oscillations (theorems on stationarity and properties of natural frequencies). Rayleigh also formulated the principle of reciprocity. By analogy with kinetic and potential energy, he introduced the dissipative function, received the name of Rayleigh and represents half the rate of energy dissipation.

In The Theory of Sound, Rayleigh also proposes an approximate method for determining the first natural frequency of a conservative system

, (16)

where
... In this case, to calculate the maximum values of potential and kinetic energies, a certain form of vibration is taken. If it coincides with the first mode of vibration of the system, we will get the exact value of the first natural frequency, but otherwise this value is always overestimated. The method gives an accuracy that is quite acceptable for practice if the static deformation of the system is taken as the first mode of vibration.

Thus, back in the 19th century, in the works of Somov and Rayleigh, a method was formed for constructing differential equations describing small oscillatory motions of discrete mechanical systems using Lagrange equations of the second kind

where in the generalized force
all force factors should be included, with the exception of elastic and dissipative ones covered by the functions R and P.

Lagrange equations (17) in matrix form, describing forced vibrations of a mechanical system, after substitution of all functions look like this

. (18)

Here Is the damping matrix, and
- column vectors, respectively, of generalized coordinates, velocities and accelerations. Common decision This equation consists of free and accompanying oscillations, which are always damped and forced oscillations occurring with the frequency of the disturbing force. We will restrict ourselves to considering only a particular solution corresponding to forced fluctuations. As excitation, Rayleigh considered generalized forces that vary according to the harmonic law. Many attributed this choice to the simplicity of the case under consideration, but Rayleigh gives a more convincing explanation - the Fourier series expansion.

Thus, for a mechanical system with more than two degrees of freedom, the solution of the system of equations presents certain difficulties, which increase like an avalanche with an increase in the order of the system. Already at five to six degrees of freedom, the problem of forced vibrations cannot be manually solved by the classical method.

In the theory of vibrations of mechanical systems, small (linear) vibrations of discrete systems played a special role. The spectral theory developed for linear systems does not even require the construction of differential equations, and to obtain a solution, one can immediately write down systems of linear algebraic equations. Although in the middle of the 19th century, methods for determining eigenvectors and eigenvalues (Jacobi), as well as solving a system of linear algebraic equations (Gauss), were developed, their practical application, even for systems with a small number of degrees of freedom, was out of the question. Therefore, before the advent of sufficiently powerful computers, many different methods were developed for solving the problem of free and forced vibrations of linear mechanical systems. Many outstanding scientists - mathematicians and mechanics - have dealt with these problems; they will be discussed below. The advent of powerful computing technology made it possible not only to solve linear problems of large dimensions in a split second, but also to automate the very process of compiling systems of equations.

Thus, during the XVIII century. in the theory of small oscillations of systems with a finite number of degrees of freedom and oscillations of continuous elastic systems, the basic physical schemes were developed and the principles essential for mathematical analysis problems. However, to create the theory of mechanical vibrations as an independent science, there was a lack of a unified approach to solving problems of dynamics, and for its more rapid development there were no technical requirements.

The growth of large-scale industry in the late 18th and early 19th centuries, caused by the widespread introduction of the steam engine, led to the separation of applied mechanics into a separate discipline. But until the end of the 19th century, strength calculations were carried out in a static setting, since the machines were still low-powered and slow-moving.

By the end of the 19th century, with the increase in speeds and the decrease in the size of machines, it became impossible to neglect the fluctuations. Numerous accidents resulting from the onset of resonance or fatigue failure during vibrations forced engineers to pay attention to vibrational processes. Of the problems that arose during this period, the following should be noted: collapse of bridges from passing trains, torsional vibrations of shafting and vibration of ship hulls, excited by the inertial forces of moving parts of unbalanced machines.

IIIperiod- the formation and development of the applied theory of oscillations (1900–1960s). Developing mechanical engineering, improvement of locomotives and ships, the emergence of steam and gas turbines, high-speed internal combustion engines, cars, airplanes, etc. demanded a more accurate analysis of stresses in machine parts. This was dictated by the requirements for a more economical use of metal. The lightweight construction has given rise to vibration problems, which are increasingly becoming critical in matters of machine strength. At the beginning of the 20th century, numerous accidents convincingly show what catastrophic consequences can be caused by neglecting vibrations or ignorance of them.

The emergence of new technology, as a rule, poses new problems for the theory of oscillations. So in the 30-40s. new problems arose, such as stall flutter and shimmy in aviation, bending and flexural-torsional vibrations of rotating shafts, etc., which required the development of new methods for calculating vibrations. In the late 1920s, first in physics, and then in mechanics, the study of nonlinear oscillations began. In connection with the development of automatic control systems and other technical requirements, starting from the 30s, the theory of motion stability has been widely developed and applied, the basis of which was A. M. Lyapunov's doctoral dissertation "The General Problem of Motion Stability".

The absence of an analytical solution for the problems of the theory of oscillations, even in a linear setting, on the one hand, and computer technology, on the other, has led to the development of a large number of various numerical methods for their solution.

The need to calculate vibrations for various types of technology led to the appearance in the 1930s of the first training courses the theory of oscillations.

Transition to IVperiod(early 1960s - present) is associated with the era of scientific and technological revolution and is characterized by the emergence of new technology, primarily aviation and space, robotic systems. In addition, the development of power engineering, transport, and others has put forward the problems of dynamic strength and reliability in the first place. This is due to an increase in operating speeds and a decrease in material consumption with a simultaneous striving to increase the service life of machines. In the theory of oscillations, more and more problems are solved in a nonlinear setting. In the field of oscillations of continuous systems, under the influence of the requirements of aviation and space technology, problems of the dynamics of plates and shells arise.

The greatest influence on the development of the theory of oscillations in this period was exerted by the emergence and rapid development of electronic computing technology, which led to the development of numerical methods for calculating oscillations.

Oscillating motion is called any movement or change of state, characterized by one or another degree of repetition in time of the values of physical quantities that determine this movement or state. Oscillations are inherent in all natural phenomena: pulsating radiation from stars; planets rotate with a high degree of periodicity Solar system; winds excite vibrations and waves on the surface of the water; inside any living organism, various, rhythmically repeating processes continuously occur, for example, the human heart beats with amazing reliability.

Oscillations stand out in physics mechanical and electromagnetic. With the help of propagating mechanical fluctuations in the density and pressure of air, which we perceive as sound, as well as very rapid fluctuations in electric and magnetic fields, which we perceive as light, we receive a large number of direct information about the world around us. Examples of oscillatory motion in mechanics are oscillations of pendulums, strings, bridges, etc.

Oscillations are called periodic, if the values of physical quantities that change in the course of oscillations are repeated at regular intervals. The simplest type of periodic vibration is harmonic vibration. Oscillations are called harmonic oscillations in which the change in the oscillating quantity over time occurs according to the sine (or cosine) law:

where x is the displacement from the equilibrium position;

A - vibration amplitude - maximum displacement from the equilibrium position;

- cyclic frequency;

- the initial phase of the oscillation;

- oscillation phase; it determines the displacement at any moment in time, i.e. determines the state of the oscillatory system.

In the case of strictly harmonic oscillations, the quantities A, and do not depend on time.

Cyclic frequency associated with the period T of oscillations and the frequency ratio:

(2)

Period T fluctuations called the smallest period of time after which the values of all physical quantities characterizing the fluctuations are repeated.

Frequency vibrations is the number of complete vibrations per unit time, measured in hertz (1 Hz = 1
).

Cyclic frequency is numerically equal to the number of oscillations performed in 2 seconds.

Oscillations arising in a system that is not subject to the action of variable external forces, as a result of any initial deviation of this system from a state of stable equilibrium, are called free(or your own).

If the system is conservative, then no energy dissipation occurs during oscillations. In this case, free vibrations are called undamped.

Speed the fluctuations of the point are defined as the derivative of the time displacement:

(3)

Acceleration of the oscillating point is equal to the derivative of the velocity with respect to time:

(4)

Equation (4) shows that the acceleration during harmonic oscillations is variable, therefore, the oscillation is due to the action of a variable force.

Newton's second law allows us to write in general terms the relationship between force F and acceleration with rectilinear harmonic vibrations material point with mass
:

where
, (6)

k - coefficient of elasticity.

Thus, the force causing harmonic vibration is proportional to the displacement and is directed against the displacement. In this regard, it is possible to give a dynamic definition of harmonic vibration: harmonic is the vibration caused by a force directly proportional to the displacement x and directed against the displacement.

The restoring force can be, for example, an elastic force. Forces that have a different nature than elastic forces, but also satisfy condition (5), are called quasi-elastic.

In the case of rectilinear vibrations along the x-axis, the acceleration equals:

Substituting this expression to speed up and the meaning of strength
into Newton's second law, we get the basic equation of rectilinear harmonic vibrations:

or
(7)

The solution to this equation is equation (1).

MINISTRY OF EDUCATION OF THE RUSSIAN FEDERATION

KABARDINO-BALKAR STATE

UNIVERSITY them. H. M. BERBEKOVA

FOUNDATIONS OF THE THEORY OF VIBRATIONS

FUNDAMENTALS OF THEORY, PROBLEMS FOR HOME TASKS,

EXAMPLES OF SOLUTIONS

For students of mechanical specialties of universities

Nalchik 2003

Reviewers:

- Doctor of Physical and Mathematical Sciences, Professor, Director of the Research Institute of Applied Mathematics and Automation of the Russian Academy of Sciences, Hon. scientist of the Russian Federation, academician of AMAN.

Doctor of Physical and Mathematical Sciences, Professor, Head of the Department of Applied Mathematics of the Kabardino-Balkarian State Agricultural Academy.

Kulterbaev theory of oscillations. Fundamentals of theory, tasks for homework, examples of solutions.

Textbook for students of higher technical educational institutions studying in the areas of training graduates 657800 - Design and technological support of machine-building industries, 655800 Food engineering. –Nalchik: Publishing house of KBSU im. , 20s.

The book outlines the foundations of the theory of oscillations of linear mechanical systems, as well as tasks for homework with examples of their solution. The theory content and assignments are focused on mechanical students.

Both discrete and distributed systems are considered. The number of mismatched options for homework allows them to be used for a large flow of students.

The publication can also be useful for teachers, graduate students and specialists in various fields of science and technology who are interested in applications of the theory of oscillations.

Foreword

The book is written based on the course, read by the author at the engineering and technical faculty of the Kabardino-Balkarian State University for students of mechanical specialties.

Mechanisms and structures modern technology they often work under complex dynamic loading conditions, therefore, the constant interest in the theory of oscillations is supported by the demands of practice. The theory of vibrations and its applications have an extensive bibliography, including a considerable number of textbooks and teaching aids. Some of them are given in the bibliography at the end of this tutorial. Almost all existing educational literature is intended for readers who study this course in large volumes and specialize in areas of engineering activity, one way or another, significantly related to the dynamics of structures. Meanwhile, at present, all engineers of mechanical specialties feel the need to master the theory of oscillations at a fairly serious level. An attempt to satisfy such requirements leads to the introduction of small-volume special courses into the educational programs of many universities. This tutorial is designed to cater to just such requests, and contains the basics of theory, homework problems, and examples for solving them. This justifies the limited volume of the textbook, the choice of its content and the title: "Foundations of the theory of vibrations." Indeed, the textbook sets out only the basic questions and methods of the discipline. The interested reader can take advantage of well-known scientific monographs and teaching aids given at the end of this publication for in-depth study theory and its many applications.

The book is intended for a reader who has training in the volume of ordinary university courses higher mathematics, theoretical mechanics and strength of materials.

In the study of such a course, a significant amount of homework takes place in the form of coursework, control, calculation and design, calculation and graphic and other work that requires a lot of time. Existing problem books and manuals for solving problems are not intended for these purposes. In addition, there is a clear expediency in combining theory and homework in one edition, united by a common content, thematic focus and complementing each other.

When completing and completing homework assignments, the student is faced with many questions that are not stated or not sufficiently explained in the theoretical part of the discipline; he has difficulties in presenting the course of solving the problem, ways of arguing for decisions made, structuring and formatting records.

Teachers are also experiencing difficulties, but already of an organizational nature. They often have to revise the volume, content and structure of homework, make up numerous options for tasks, ensure the timely issuance of mismatched tasks on a massive scale, conduct numerous consultations, explanations, etc.

This manual is intended, inter alia, to reduce and eliminate difficulties and difficulties of this nature in the context of mass education. It contains two tasks, according to their topics, covering the most important and basic questions of the course:

1. Oscillations of systems with one degree of freedom.

2. Oscillations of systems with two degrees of freedom.

In terms of their volume and content, these tasks can become design and design work for full-time, part-time and part-time students or tests for students. extramural form learning.

For the convenience of readers, the book uses autonomous numbering of formulas (equations) and figures within each paragraph using the usual decimal number in brackets. References within the current paragraph are made simply by specifying such a number. If it is necessary to refer to the formula of the previous paragraphs, indicate the number of the paragraph and then through a dot - the number of the formula itself. So, for example, the notation (3.2.4) corresponds to the formula (4) in paragraph 3.2 of this chapter. The reference to the formula of the previous chapters is made in the same way, but with the indication in the first place of the chapter number and period.

The book is an attempt to satisfy requests vocational training students of certain directions. The author is aware that it, apparently, will not be free from shortcomings, and therefore will accept with gratitude possible criticism and comments from readers for improving subsequent editions.

The book may also be useful to specialists interested in applications of the theory of oscillations in different areas physics, engineering, construction and other areas of knowledge and production activities.

ChapterI

INTRODUCTION

1.The subject of the theory of vibrations

A certain system moves in space so that its state at each moment of time t is described by a certain set of parameters: https://pandia.ru/text/78/502/images/image004_140.gif "width =" 31 "height =" 23 src = ">. gif" width = "48" height = "24"> and external influences. And then the task is to predict the further evolution of the system in time: (Fig. 1).

Let one of the changing characteristics of the system be,. There can be various characteristic varieties of its change in time: monotonic (Fig. 2), non-monotonic (Fig. 3), substantially non-monotonic (Fig. 4).

The process of changing a parameter, which is characterized by multiple alternate increase and decrease of the parameter in time, is called oscillatory process or simply fluctuations. Oscillations are widespread in nature, technology and human activity: the rhythms of the brain, pendulum oscillations, heartbeats, oscillations of stars, oscillations of atoms and molecules, oscillations of current in an electric circuit, oscillations of air temperature, fluctuations in food prices, vibration of sound, vibration strings of a musical instrument.

The subject of this course is mechanical vibrations ie, vibrations in mechanical systems.

2. Classification of oscillatory systems

Let be u(NS, t) is the state vector of the system, f(NS, t) is the vector of actions on the system from the side environment(fig. 1). The dynamics of the system is described by the operator equation

L u(NS, t) = f(NS, t), (1)

where the operator L is given by the equations of oscillations and additional conditions(boundary, initial). In such an equation, u and f can also be scalars.

The simplest classification of oscillatory systems can be made by their number of degrees of freedom... The number of degrees of freedom is the number of independent numerical parameters that uniquely determine the configuration of the system at any time moment t. On this basis, oscillatory systems can be attributed to one of three classes:

1)Systems with one degree of freedom.

2)Systems with a finite number of degrees of freedom... They are often referred to as discrete systems.

3)Systems with an infinite uncountable number of degrees of freedom (continuous, distributed systems).

In fig. 2 shows a number of illustrative examples for each of their classes. For each scheme, the number of degrees of freedom is indicated in circles. The last diagram shows a distributed system in the form of an elastic deformable beam. To describe its configuration, a function u (x, t) is required, that is, an infinite set of values of u.

Each class of oscillatory systems has its own mathematical model. For example, a system with one degree of freedom is described by an ordinary differential equation of the second order, systems with a finite number of degrees of freedom - by a system of ordinary differential equations, distributed systems - by partial differential equations.

Depending on the type of operator L in model (1), oscillatory systems are divided into linear and non-linear... The system is considered linear if the corresponding operator is linear, i.e., satisfies the condition

https://pandia.ru/text/78/502/images/image014_61.gif "width =" 20 height = 24 "height =" 24 ">. jpg" width = "569" height = "97">
For linear systems, it is true superposition principle(the principle of independence of action of forces). Its essence is based on an example (fig..gif "width =" 36 "height =" 24 src = "> is as follows..gif" width = "39" height = "24 src ="> .. gif "width =" 88 "height =" 24 ">.

Stationary and non-stationary systems. Have stationary systems on the considered period of time, the properties do not change in time. Otherwise the system is called non-stationary. The next two figures clearly demonstrate the fluctuations in such systems. In fig. 4 shows oscillations in a stationary system under steady-state conditions, in Fig. 5 - oscillations in a non-stationary system.

Processes in stationary systems are described by differential equations with coefficients constant in time, in non-stationary systems - with variable coefficients.

Autonomous and non-autonomous systems. V autonomous systems external influences are absent. Oscillatory processes in them can occur only due to internal energy sources or due to the energy imparted to the system at the initial moment of time. In operator equation (1), then the right-hand side does not depend on time, i.e. f(x, t) = f(x). The rest of the systems are non-autonomous.

Conservative and non-conservative systems. https://pandia.ru/text/78/502/images/image026_20.jpg "align =" left hspace = 12 "width =" 144 "height =" 55 "> Free vibrations. Free vibrations are performed in the absence of variable external influence, without an influx of energy from the outside. Such fluctuations can occur only in autonomous systems (Fig. 1).

Forced vibrations. Such fluctuations take place in non-autonomous systems, and their sources are variable external influences (Fig. 2).

Parametric vibrations. The parameters of an oscillatory system can change over time, and this can become a source of oscillations. Such vibrations are called parametric. The upper point of suspension of the physical pendulum (fig..gif "width =" 28 "height =" 23 src = ">, which is the cause of lateral parametric oscillations (fig. 5).

Self-oscillations(self-excited oscillations). For such oscillations, the sources have a non-oscillatory nature, and the sources themselves are included in the oscillatory system. In fig. 6 shows a spring-loaded mass lying on a moving belt. Two forces act on it: the friction force and the elastic force of the spring tension, and they change over time. The first depends on the difference between the speeds of the tape and the mass, the second on the magnitude and sign of the deformation of the spring, therefore the mass is under the influence of the resultant force, directed now to the left, now to the right, and oscillates.

In the second example (Fig. 7), the left end of the spring moves to the right at a constant speed v, as a result of which the spring moves the load along the stationary surface. A situation similar to that described for the previous case is formed, and the load begins to oscillate.

4. Kinematics of periodic oscillatory processes

Let the process be characterized by one scalar variable, which is, for example, displacement. Then - speed, - acceleration .. gif "width =" 11 height = 17 "height =" 17 "> the condition

then the vibrations are called periodic(fig. 1). Moreover, the smallest of these numbers is called period of fluctuations... The unit of measurement for the period of oscillation is, most often, a second, denoted with or sec. Also used are units of measurement in minutes, hours, etc. Another, also important characteristic of a periodic oscillatory process is vibration frequency

determining quantity full cycles fluctuations per 1 unit of time (for example, per second). This frequency is measured in or hertz (Hz), so that means 5 complete cycles of vibration in one second. In mathematical calculations of the theory of oscillations, it turns out to be more convenient angular frequency

measured in https://pandia.ru/text/78/502/images/image041_25.gif "width =" 115 height = 24 "height =" 24 ">.

The simplest of the periodic oscillations, but extremely important for building a theoretical basis for the theory of oscillations, are harmonic (sinusoidal) oscillations that vary according to the law

https://pandia.ru/text/78/502/images/image043_22.gif "width =" 17 "height =" 17 src = "> - amplitude, - oscillation phase, - initial phase..gif" width = " 196 "height =" 24 ">,

and then acceleration

Instead of (1), an alternative notation is often used

https://pandia.ru/text/78/502/images/image050_19.gif "width =" 80 "height =" 21 src = ">. Descriptions (1) and (2) can be presented in the form

Between the constants in formulas (1), (2), (3) there are easily proved relations

The use of methods and representations of the theory of functions of complex variables greatly simplifies the description of oscillations. In this case, the central place is occupied by Euler's formula

Here https://pandia.ru/text/78/502/images/image059_15.gif "width =" 111 "height =" 28 ">. (4)

Formulas (1) and (2) are contained in (4). For example, sinusoidal oscillations (1) can be represented as an imaginary component (4)

and (2) - in the form of a real component

Polyharmonic vibrations. The sum of two harmonic vibrations with the same frequencies will be a harmonic vibration with the same frequency

The terms could be with unequal frequencies

Then the sum (5) will be a periodic function with a period, only if,, where and are integers, and an irreducible fraction, rational number... In general, if two or more harmonic oscillations have frequencies with ratios in the form rational fractions, then their sums are periodic, but not harmonic oscillations. Such vibrations are called polyharmonic.

If the periodic oscillations are not harmonic, then it is still often advantageous to represent them as a sum of harmonic oscillations using Fourier series

Here https://pandia.ru/text/78/502/images/image074_14.gif "width =" 15 "height =" 19 "> is the harmonic number, characterizes the mean deviation, https://pandia.ru/text /78/502/images/image077_14.gif "width =" 139 height = 24 "height =" 24 "> - the first, fundamental harmonic, (https://pandia.ru/text/78/502/images/image080_11. gif "width =" 207 "height =" 24 "> forms frequency spectrum hesitation.

Note. The Dirichlet theorem for a periodic function serves as a theoretical substantiation of the possibility of representing a function of an oscillatory process by a Fourier series:

If a function is set on a segment and is piecewise continuous, piecewise monotone and bounded on it, then its Fourier series converges at all points of the segment https://pandia.ru/text/78/502/images/image029_34.gif "width = "28" height = "23 src ="> is the sum of the trigonometric Fourier series of the function f (t), then at all points of continuity of this function

and at all break points

Besides,

Obviously, real oscillatory processes satisfy the conditions of the Dirichlet theorem.

In the frequency spectrum, each frequency corresponds to the amplitude Аk and the initial phase https://pandia.ru/text/78/502/images/image087_12.gif "width =" 125 "height =" 33 ">, .

They form amplitude spectrum https://pandia.ru/text/78/502/images/image090_9.gif "width =" 35 "height =" 24 ">. A visual representation of the amplitude spectrum is given in Fig. 2.

Determination of the spectrum of frequencies and Fourier coefficients is called spectral analysis... From the theory of Fourier series, the formulas are known