Fundamentals of the theory of oscillations. Classification of oscillatory systems

Course program theory of vibrations for students 4 FACI course

The discipline is based on the results of such disciplines as classical general algebra, the theory of ordinary differential equations, theoretical mechanics, and the theory of functions of a complex variable. A feature of studying the discipline is frequent use of the apparatus mathematical analysis and other related mathematical disciplines, practical use important examples from subject area theoretical mechanics, physics, electrical engineering, acoustics.

1. Qualitative analysis of motion in a conservative system with one degree of freedom

• Phase plane method
• Dependence of the oscillation period on the amplitude. Soft and hard systems

2. Duffing equation

• Expression for the general solution of the Duffing equation in elliptic functions

3. Quasilinear systems

• Van der Pol Variables
• Averaging method

4. Relaxation oscillations

• Van der Pol equation
• Singularly perturbed systems of differential equations

5. Dynamics of nonlinear autonomous systems general view with one degree of freedom

• The concept of “roughness” of a dynamic system
• Bifurcations of dynamic systems

6. Elements of Floquet's theory

• Normal solutions and multipliers linear systems differential equations with periodic coefficients
• Parametric resonance

7. Hill's equation

• Analysis of the behavior of solutions to a Hill type equation as an illustration of the application of Floquet theory to linear Hamiltonian systems with periodic coefficients
• Mathieu's equation as special case Hill type equations. Ines-Strett diagram

8. Forced oscillations in a system with a nonlinear restoring force

• Relationship between the amplitude of oscillations and the magnitude of the driving force applied to the system
• Changing the driving mode when changing the frequency of the driving force. The concept of “dynamic” hysteresis

9. Adiabatic invariants

• Action-Angle Variables
• Conservation of adiabatic invariants under qualitative change nature of movement

10. Dynamics of multidimensional dynamical systems

• The concept of ergodicity and mixing in dynamic systems
• Poincaré map

11. Lorentz equations. Strange attractor

• Lorentz equations as a model of thermoconvection
• Bifurcations of solutions to Lorentz equations. Transition to chaos
• Fractal structure of a strange attractor

12. One-dimensional displays. Feigenbaum's versatility

• Quadratic mapping - the simplest nonlinear mapping
• Periodic orbits of mappings. Bifurcations of periodic orbits

Literature (main)

1. Moiseev N.N. Asymptotic methods of nonlinear mechanics. – M.: Nauka, 1981.

2. Rabinovich M.I., Trubetskov D.I. Introduction to the theory of oscillations and waves. Ed. 2nd. Research Center “Regular and Chaotic Dynamics”, 2000.

3. Bogolyubov N.N., Mitropolsky Yu.A. Asymptotic methods in the theory of nonlinear oscillations. – M.: Nauka, 1974.

4. Butenin N.V., Neimark Yu.I., Fufaev N.A. Introduction to the theory of nonlinear oscillations. – M.: Nauka, 1987.

5. Loskutov A.Yu., Mikhailov A.S. Introduction to synergetics. – M.: Nauka, 1990.

6. Karlov N.V., Kirichenko N.A. Oscillations, waves, structures.. - M.: Fizmatlit, 2003.

Literature (additional)

7. Zhuravlev V.F., Klimov D.M. Applied methods in the theory of oscillations. Publishing house "Science", 1988.

8. Stocker J. Nonlinear oscillations in mechanical and electrical systems. – M.: Foreign literature, 1952.

9. Starzhinsky V.M., Applied methods of nonlinear oscillations. – M.: Nauka, 1977.

10. Hayashi T. Nonlinear oscillations in physical systems. – M.: Mir, 1968.

11. Andronov A.A., Witt A.A., Khaikin S.E. Theory of oscillations. – M.: Fizmatgiz, 1959.

The concept of oscillations. Let's consider a certain system, i.e. a set of objects interacting with each other and with the environment according to a certain law. It can be like a mechanical system material points, absolutely solids, elastic and generally deformable bodies, etc., as well as electrical, biological and mixed (for example, electromechanical) systems. Let the state of the system at each moment of time be described by a certain set of parameters. The task of the theory is to predict the evolution of a system over time, given the initial state of the system and the external influence on it.

Let's take one of the numerical parameters of the system, denoting it by and. It could be scalar quantity, one of the components of a vector or tensor, etc. Let us consider the change in this parameter over a certain period of time, for example, at This change can be monotonic, non-monotonic, essentially non-monotonic (Fig. 1). The last case is of greatest interest.

The process of changing a parameter, which is characterized by multiple alternating increases and decreases of the parameter over time, is called an oscillatory process or simply oscillations, and the corresponding parameter is called an oscillating value.

It is impossible to establish a clear boundary separating oscillatory processes from non-oscillating. For example, in economics a process of the type shown in Fig. 1b can be attributed to oscillatory processes. It is possible to formulate more general definition oscillatory process: the parameter performs on given segment the time of oscillation relative to the parameter (and vice versa), if the difference in this segment changes sign many times (Fig. 1d). For example, we can talk about an oscillatory change in the angle of rotation of the disk relative to uniform rotation with a constant angular velocity

If all or the most essential parameters of a system are oscillating quantities, then the system is said to be experiencing oscillations. A system capable of oscillating under certain conditions is called an oscillatory system. Strictly speaking, any system fits this definition, since for any system it is possible to choose such an impact under which it will perform oscillatory motion. Therefore, they usually use more narrow definition: a system is called oscillatory if it is capable of oscillating in the absence of external influences (only due to the initially accumulated energy).

The place of oscillatory processes in science and technology. Most of the processes observed in nature and technology are oscillatory. Oscillatory processes include a wide variety of phenomena: from brain rhythms and heartbeats to vibrations of stars, nebulae and others. space objects; from the vibrations of atoms or molecules in a solid to climate changes on Earth, from the vibrations of a sounding string to earthquakes. All acoustic and propagation phenomena electromagnetic waves, are also accompanied by oscillatory processes.

Rice. I. Change of parameter: a - monotonic; b - non-monotonic; c - essentially non-monotonic; r - relative change in parameters

IN this volume Mainly mechanical systems will be considered. The oscillatory processes occurring in these systems are called mechanical oscillations. In technology, especially in mechanical engineering, the term vibration is also widely used. It is almost synonymous with the terms mechanical vibrations or vibrations of a mechanical system. The term vibration is most often used where the vibrations have a relatively small amplitude and not too low a frequency (for example, one can hardly accept the term vibration when talking about the oscillations of a clock pendulum or the swinging of a swing).

Applied theory of vibrations and vibration engineering. The set of methods and means for measuring quantities characterizing vibrations is called vibrometry. A set of methods and means to reduce harmful effects vibrations on people, devices and mechanisms is called vibration protection. A set of technological techniques based on the targeted use of vibration is called vibration processing, and the use of vibration to move materials, products, etc. is called vibration transportation. To ensure the ability of objects to perform their functions and maintain parameters within limits established standards, and also to maintain strength under vibration conditions, calculations for vibration resistance and vibration strength or, in a more general formulation, for vibration reliability are required. The purpose of vibration testing is to study the vibration resistance, vibration strength and efficiency of objects under vibration conditions, as well as to study the effectiveness of vibration protection; The task of vibration diagnostics is to study the state of an object based on the analysis of operational or artificially excited vibrations.

Development modern technology poses a wide variety of tasks for engineers related to the calculation of various structures, the design, production and operation of all kinds of machines and mechanisms.

The study of the behavior of any mechanical system always begins with the choice of a physical model. When moving from a real system to its physical model, one usually simplifies the system, neglecting factors that are unimportant for a given problem. Thus, when studying a system consisting of a load suspended on a thread, the size of the load, the mass and compliance of the thread, the resistance of the medium, friction at the point of suspension, etc. are neglected; this produces a well-known physical model - a mathematical pendulum.

Limitation physical models plays a significant role in research oscillatory phenomena in mechanical systems.

Physical models that are described by systems of linear differential equations with constant coefficients are usually called linear.

Selection linear models into a special class is called for a number of reasons:

Using linear models, a wide range of phenomena occurring in different mechanical systems Oh;

Integrating linear differential equations with constant coefficients is, from a mathematical point of view, an elementary task and therefore the research engineer strives to describe the behavior of the system using a linear model whenever possible.

Basic concepts and definitions

Oscillations of a system are considered small if deviations and velocities can be considered as quantities of the first order of smallness compared to the characteristic sizes and velocities of points in the system.

A mechanical system can perform small oscillations only near a stable equilibrium position. The equilibrium of the system can be stable, unstable and indifferent (Fig. 3. 8).

Rice. 3.8 Different kinds equilibrium

The equilibrium position of a system is stable if the system whose equilibrium is disturbed by a very small initial deviation and/or small initial speed, makes a movement around this position.

Criterion for the stability of the equilibrium position of conservative systems with holonomic and fixed connections is established by the type of dependence of the potential energy of the system on generalized coordinates. For a conservative system c
degrees of freedom, the equilibrium equations have the form

, i.e.
, Where
.

The equilibrium equations themselves do not make it possible to assess the nature of the stability or instability of the equilibrium position. It only follows from them that the equilibrium position corresponds to an extreme value of potential energy.

The stability condition for the equilibrium position (sufficient) is established by the Lagrange–Dirichlet theorem:

if in the equilibrium position of the system potential energy has a minimum, then this situation is stable.

The condition for the minimum of any function is that its second derivative is positive when the first derivative is equal to zero. That's why

.

If the second derivative is also zero, then to assess stability it is necessary to calculate successive derivatives

,

and if the first one doesn't equal to zero derivative has an even order and is positive, then the potential energy at
has a minimum, and therefore this equilibrium position of the system is stable. If this derivative has an odd order, then when
there is no maximum or minimum. An assessment of the state of equilibrium of a system in a position where it does not have a minimum potential energy is given in special theorems by A. M. Lyapunov.

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

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

UDC 534.01
X-35
Fundamentals of the theory of vibrations of mechanical systems / V.K. Khegai,
D.N. Levitsky, O.N. Kharin, A.S. Popov. – Ukhta: USTU, 2002. – 108 p.
ISBN 5-88179-285-8
The textbook examines the fundamentals of the theory of vibrations of mechanical systems, which are based on general course theoretical mechanics. Special attention devoted to the application of Lagrange's second equations
row. The manual consists of six chapters, each of which is devoted to a specific type of oscillation. One chapter is devoted to the fundamentals of the theory of stability of motion and equilibrium of mechanical systems.
For better development 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,
may also be useful for students of other specialties.
Reviewers: Department of Theoretical Mechanics of St. Petersburg
State Forestry Academy (head of the department, Doctor of Technical Sciences, Professor Yu.A. Dobrynin); Head of the integrated drilling department of SeverNIPIGaz, Ph.D., Associate Professor Yu.M. Gerzhberg.

© Ukhta State Technical University, 2002
©Khegai V.K., Levitsky D.N., Kharin O.N., Popov A.S., 2002
ISBN 5-88179-285-8

3
Table of contents
Preface........................................................ ........................................................ .................. 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................................................... ............................................ 8
1.4. Langrange's equation................................................... ................................................ 9
1.5. Examples for composing Langrange equations of the second kind.................................... 11
Chapter II. Stability of motion and equilibrium of conservative systems......... 20
2.1. Introduction........................................................ ........................................................ ................... 20
2.2. Lyapunov functions. Sylvester's criterion................................................... ............. 21
2.3. Equation of perturbed motion................................................... ........................... 23
2.4. Lyapunov's theorem on stability of motion.................................................... .......... 26
2.5. Lagrange's theorem on the stability of equilibrium
conservative system................................................... ........................................................ 29
2.6. Stability of equilibrium of a conservative system with one
degrees of freedom........................................................ ........................................................ ........... thirty
2.7. Examples of stability of equilibrium of a conservative system.................................... 31
Chapter III. Free oscillations of a system with one degree of freedom.................................... 39
3.1. Free oscillations 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 speed......................................................... 42
3.3. Examples of free oscillations 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. Resonance phenomenon................................................... ........................................................ .... 63
4.3. The phenomenon of beating................................................... ........................................................ ........ 66
4.4. Dynamic coefficient................................................... .................................... 68
4.5. Examples on forced oscillations systems
with one degree of freedom......................................................... ................................................... 70
Chapter V. Free vibrations 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 common decision........................................................................ 78
5.2. Own forms.................................................................................................. 80
5.3. Examples of free oscillation 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
Bibliography................................................................ ........................................... 107

4
Preface
On modern stage development high school Problematic and research forms training.
Dynamic processes in machines and mechanisms are of decisive importance both for calculations at the design stage of new structures and for determining technological modes during operation. It is difficult to name an area 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 discusses some individual issues from theory
vibrations and stability of mechanical systems. Theoretical information
explained with examples.
The main purpose of this methodological manual− link
area of ​​applications of theoretical and analytical mechanics with problems
special departments that train mechanical engineers.

5
Chapter I. BRIEF INFORMATION FROM THE 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 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 stable position
equilibrium, the 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 generalized coordinates, we expand the expression of potential energy P into the Maclaurin series in powers of qj

∂П
1 S S ∂2 P
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 potential energy is determined with accuracy
up to some additive constant, the potential energy at 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 when the system of forces is in equilibrium

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

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

⎛ ∂P
⎜⎜
⎝ ∂q j

⎞
⎟⎟ = 0
⎠0

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

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

⎞
⎟⎟ qi q j .
⎠0

Let's denote

⎛ ∂2 P
⎜⎜
⎝ ∂qi ∂q j

⎞
⎟⎟ = cij = c ji ,
⎠0

Where cij are generalized stiffness coefficients.
The final expression for potential energy is

1 S S
П = ∑∑cij qi q j .
2 i =1 j =1

From (1.9.) it is clear 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

1n
T = ∑mk vk2 ,
2 k =1

Where mk and vк are the mass and speed of the kth point of the system.
When moving 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 kth point of the system.
−

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

V k its value
_

∂rk
∂q1

∂rk
∂q2

∂rk
∂qs

Then the expression for kinetic energy (1.10) will take 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 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 small deviations of the system from the position are considered
equilibrium, then in equality (1.14) we restrict ourselves to only the first constant terms

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

Aij = (Aij)0 = aij

Then the expression for kinetic energy (1.13) will take 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

1 S
T= ∑
2 i=1

Constants aij are generalized inertia coefficients.
From (1.16) it is clear that the kinetic energy of the system T is homogeneous
quadratic function of generalized speeds.

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

Let us assume that the resistance forces R k (k = 1, 2,..., n) acting
to 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 resistance forces for a 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 defined by the formula
n

Then the generalized resistance forces are determined by the formulas

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

The dissipative function, by analogy with the kinetic energy of the system, 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 generalized dissipation coefficients.
1.4. Lagrange equation of the second kind
The position of a holonomic system having 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
dynamics equation
S

Q иj)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 jth 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 for δ q j separately will be equal to zero, i.e.

Q j + Qиj = 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 definition of generalized force, we have

Q иj = ∑ 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
– inertia force at 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
∂qs
_

⎛ _
d ⎜ ∂ rk
dt ⎜ ∂q j
⎝

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

Substituting 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), we rewrite expression (1.25) in the form

⎡ ⎛
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 potential, then
for generalized forces the 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

MINISTRY OF EDUCATION OF THE RUSSIAN FEDERATION

KABARDINO-BALKARIAN STATE

UNIVERSITY named after. Kh. M. BERBEKOVA

FUNDAMENTALS OF THE THEORY OF OSCILLATIONS

BASICS OF THEORY, TASKS FOR HOMEWORK,

EXAMPLES OF SOLUTIONS

For university students of mechanical specialties

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, Honored. 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. Basic theory, homework problems, examples of solutions.

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

The book outlines the fundamentals of the theory of oscillations of linear mechanical systems, and also provides homework problems with examples of their solutions. The content of the theory and assignments are aimed at students of mechanical specialties.

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 may 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.

© Kabardino-Balkarian State University them.

Preface

The book is based on a course read by the author at the Faculty of Engineering and Technology of the Kabardino-Balkarian State University for students of mechanical specialties.

The mechanisms and structures of modern technology often operate under complex dynamic loading conditions, so the constant interest in the theory of vibrations is supported by practical needs. The theory of oscillations 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 manual. Almost all existing educational literature is intended for readers studying this course in large volume and specializing in areas engineering activities, one way or another, significantly related to the dynamics of structures. Meanwhile, at present, all mechanical engineers feel the need to master the theory of vibrations at a fairly serious level. An attempt to satisfy such requirements leads to the introduction of small-sized universities into the educational programs of many universities. special courses. This textbook is designed to satisfy just such requests, and contains the basics of theory, homework problems and examples of how to solve them. This justifies the limited volume of the textbook, the choice of its content and the title: “Fundamentals of the Theory of Oscillations.” Indeed, the textbook outlines only the basic issues and methods of the discipline. The interested reader can take advantage of well-known scientific monographs and teaching aids given at the end this edition, For in-depth study theory and its many applications.

The book is intended for a reader who has training in the scope of regular college courses. higher mathematics, theoretical mechanics and strength of materials.

In the study of such a course, a significant amount is occupied by homework in the form of coursework, tests, calculation and design, calculation and graphic and other works that require quite a lot of time. Existing problem books and problem solving aids are not intended for these purposes. In addition, there is a clear advisability in combining theory and homework in one publication, combined general content, thematic focus and complementary.

When completing and completing homework, the student is faced with many questions that are not stated or insufficiently explained in the theoretical part of the discipline; he has difficulties in describing the progress of solving a problem, ways of justifying decisions made, structuring and writing notes.

Teachers are also experiencing difficulties, but of an organizational nature. They have to frequently revise the volume, content and structure of homework, draw up numerous versions of tasks, ensure timely delivery of mismatched tasks en masse, conduct numerous consultations, clarifications, etc.

This manual is intended, among other things, to reduce and eliminate the difficulties and difficulties of the listed nature in the conditions of mass education. It contains two tasks, covering the most important and basic issues of the course:

1. Oscillations of systems with one degree of freedom.

2. Oscillations of systems with two degrees of freedom.

These tasks, in their scope and content, can become calculation and design work for full-time students, part-time forms training or tests for students correspondence form training.

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. A reference within the current paragraph is made by simply indicating such a number. If it is necessary to refer to the formula of previous paragraphs, indicate the number of the paragraph and then, separated by a dot, the number of the formula itself. So, for example, notation (3.2.4) corresponds to formula (4) in paragraph 3.2 of this chapter. The reference to the formula of previous chapters is made in the same way, but with the chapter number and point indicated in the first place.

The book is an attempt to satisfy the needs vocational training students of certain directions. The author is aware that it, apparently, will not be free from shortcomings, and therefore will gratefully accept possible criticism and comments from readers to improve subsequent editions.

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

ChapterI

INTRODUCTION

1. Subject of vibration theory

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 further evolution systems in time: (Fig. 1).

Let one of the changing characteristics of the system be , . May be different characteristic varieties its changes over time: monotonic (Fig. 2), non-monotonic (Fig. 3), significantly non-monotonic (Fig. 4).

The process of changing a parameter, which is characterized by multiple alternating increases and decreases of the parameter over time, is called oscillatory process or simply fluctuations. Oscillations are widespread in nature, technology and human activity: rhythms of the brain, oscillations of a pendulum, beating of the heart, vibrations of stars, vibrations of atoms and molecules, fluctuations in current strength electrical circuit, fluctuations in air temperature, fluctuations in food prices, vibration of sound, vibration of the strings of a musical instrument.

Subject of study this course are mechanical vibrations, i.e. vibrations in mechanical systems.

2. Classification oscillatory systems

Let u(X, t) – system state vector, f(X, t) – vector of influences on the system from outside environment(Fig. 1). The dynamics of the system are described by the operator equation

L u(X, t) = f(X, 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 scalar quantities.

Most simple classification oscillatory systems can be produced according to 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 t. Based on this feature, oscillatory systems can be classified into one of three classes:

1)Systems with one degree of freedom.

2)Systems with finite number degrees of freedom. They are often also called discrete systems.

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

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

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

Depending on the type of operator L in model (1), oscillatory systems are divided into linear and nonlinear. The system is considered linear, if the operator corresponding to it 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">
Valid for linear systems superposition principle(the principle of independence of the action of forces). The essence of it using 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. U stationary systems on the considered period of time, properties do not change over time. IN otherwise system is called non-stationary. The next two figures clearly demonstrate the oscillations in such systems. In Fig. Figure 4 shows oscillations in a stationary system in steady state, 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. IN autonomous systems there are no external influences. Oscillatory processes in them can occur only due to internal sources energy or due to the energy imparted to the system in starting moment time. In operator equation (1), then the right-hand side does not depend on time, i.e. f(x, t) = f(x). The remaining 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 oscillations can only occur 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 oscillations. The parameters of the oscillatory system can change over time, and this can become a source of oscillations. Such oscillations are called parametric. Top suspension point physical pendulum(Fig..gif" width="28" height="23 src=">, which causes transverse parametric vibrations to occur (Fig. 5).

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

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

,

then the oscillations are called periodic(Fig. 1). In this case, the smallest of such numbers is called period of oscillation. The unit of measurement for the period of oscillation is most often the second, denoted s or sec. Other units of measurement are used in minutes, hours, etc. Another, also important characteristic of the periodic oscillatory process is oscillation frequency

quantifying full cycles oscillations per 1 unit of time (for example, per second). This frequency is measured in Hertz (Hz), so it means 5 complete cycles of oscillation 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 periodic oscillations, but extremely important for the construction theoretical basis theory of oscillations are harmonic (sinusoidal) oscillations, changing 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 also be presented in the form

There are easily provable relationships between the constants in formulas (1), (2), (3)

The use of methods and concepts of the theory of functions of complex variables greatly simplifies the description of oscillations. Central location in this case it takes 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 oscillations. Sum of two harmonic vibrations with the same frequencies will be a harmonic oscillation with the same frequency

The terms could have different frequencies

Then the sum (5) will be periodic function with period , only if , , where and are integers, and 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 oscillations are called polyharmonic.

If periodic oscillations not harmonic, 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, characterizing the average value of deviations, 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. Theoretical justification The possibility of representing a function of an oscillatory process with a Fourier series is the Dirichlet theorem for a periodic function:

If a function is given on a segment and is piecewise continuous, piecewise monotonic 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="> – amount trigonometric series Fourier function f(t), then at all points of continuity of this function

and at all points of discontinuity

.

Besides,

.

It is obvious that real oscillatory processes satisfy the conditions of the Dirichlet theorem.

In the frequency spectrum, each frequency corresponds to the amplitude Ak 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">. Visual representation about amplitude spectrum gives rice 2.

Determining the frequency spectrum and Fourier coefficients is called spectral analysis . From the theory of Fourier series the following formulas are known: