Harmonic oscillation is a phenomenon of periodic change of any quantity, in which the dependence on the argument has the character of a sine or cosine function. For example, a quantity oscillates harmoniously and changes over time as follows:
where x is the value of the changing quantity, t is time, the remaining parameters are constant: A is the amplitude of oscillations, ω is the cyclic frequency of oscillations, is the full phase of oscillations, is the initial phase of oscillations.
Generalized harmonic oscillation in differential form
(Any non-trivial solution to this differential equation is a harmonic oscillation with a cyclic frequency)
Types of vibrations
Free vibrations occur under the influence of internal forces of the system after the system has been removed from its equilibrium position. For free oscillations to be harmonic, it is necessary that the oscillatory system be linear (described by linear equations of motion), and there is no energy dissipation in it (the latter would cause attenuation).
Forced vibrations occur under the influence of an external periodic force. For them to be harmonic, it is enough that the oscillatory system is linear (described by linear equations of motion), and the external force itself changes over time as a harmonic oscillation (that is, that the time dependence of this force is sinusoidal).
Harmonic Equation
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Equation (1)
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gives the dependence of the fluctuating value S on time t; this is the equation of free harmonic oscillations in explicit form. However, usually the vibration equation is understood as another representation of this equation, in differential form. For definiteness, let us take equation (1) in the form
Let's differentiate it twice with respect to time:
It can be seen that the following relationship holds:
which is called the equation of free harmonic oscillations (in differential form). Equation (1) is a solution to differential equation (2). Since equation (2) is a second-order differential equation, two initial conditions are necessary to obtain a complete solution (that is, determining the constants A and included in equation (1); for example, the position and speed of the oscillatory system at t = 0.
A mathematical pendulum is an oscillator, which is a mechanical system consisting of a material point located on a weightless inextensible thread or on a weightless rod in a uniform field of gravitational forces. The period of small natural oscillations of a mathematical pendulum of length l, motionlessly suspended in a uniform gravitational field with free fall acceleration g, is equal to
and does not depend on the amplitude and mass of the pendulum.
A physical pendulum is an oscillator, which is a solid body that oscillates in a field of any forces relative to a point that is not the center of mass of this body, or a fixed axis perpendicular to the direction of action of the forces and not passing through the center of mass of this body.
Maximum speed and acceleration values
Having analyzed the equations of dependence v(t) and a(t), we can guess that speed and acceleration take maximum values in the case when the trigonometric factor is equal to 1 or -1. Determined by the formula
How to get dependencies v(t) and a(t)
7. Free vibrations. Speed, acceleration and energy of oscillatory motion. Addition of vibrations
Free vibrations(or natural vibrations) are oscillations of an oscillatory system that occur only due to the initially imparted energy (potential or kinetic) in the absence of external influences.
Potential or kinetic energy can be imparted, for example, in mechanical systems through initial displacement or initial velocity.
Freely oscillating bodies always interact with other bodies and together with them form a system of bodies called oscillatory system.
For example, a spring, a ball and a vertical post to which the upper end of the spring is attached (see figure below) are included in the oscillatory system. Here the ball slides freely along the string (friction forces are negligible). If you move the ball to the right and leave it to itself, it will oscillate freely around the equilibrium position (point ABOUT) due to the action of the elastic force of the spring directed towards the equilibrium position.
Another classic example of a mechanical oscillatory system is a mathematical pendulum (see figure below). In this case, the ball performs free oscillations under the influence of two forces: gravity and the elastic force of the thread (the Earth is also included in the oscillatory system). Their resultant is directed towards the equilibrium position.
The forces acting between the bodies of the oscillatory system are called internal forces. By external forces are called forces acting on a system from bodies outside of it. From this point of view, free oscillations can be defined as oscillations in a system under the influence of internal forces after the system is removed from its equilibrium position.
The conditions for the occurrence of free oscillations are:
1) the emergence in them of a force that returns the system to a position of stable equilibrium after it has been removed from this state;
2) lack of friction in the system.
Dynamics of free vibrations.
Body vibrations under the influence of elastic forces. Equation of oscillatory motion of a body under the action of elastic force F(see figure) can be obtained taking into account Newton's second law ( F = ma) and Hooke's law ( F control= -kx), Where m is the mass of the ball, and is the acceleration acquired by the ball under the action of elastic force, k- spring stiffness coefficient, X- displacement of the body from the equilibrium position (both equations are written in projection onto the horizontal axis Oh). Equating the right-hand sides of these equations and taking into account that the acceleration A is the second derivative of the coordinate X(displacement), we get:
.
This is the differential equation of motion of a body oscillating under the action of an elastic force: the second derivative of the coordinate with respect to time (body acceleration) is directly proportional to its coordinate, taken with the opposite sign.
Oscillations of a mathematical pendulum. To obtain the equation of oscillation of a mathematical pendulum (figure), it is necessary to expand the force of gravity F T= mg to normal Fn(directed along the thread) and tangential F τ(tangent to the trajectory of the ball - circle) components. Normal component of gravity Fn and the elastic force of the thread Fynp in total impart to the pendulum centripetal acceleration, which does not affect the magnitude of the speed, but only changes its direction, and the tangential component F τ is the force that returns the ball to its equilibrium position and causes it to perform oscillatory movements. Using, as in the previous case, Newton's law for tangential acceleration ma τ = F τ and given that F τ= -mg sinα, we get:
a τ= -g sinα,
The minus sign appeared because the force and angle of deviation from the equilibrium position α have opposite signs. For small deflection angles sin α ≈ α. In turn, α = s/l, Where s- arc O.A., I- thread length. Considering that and τ= s", we finally get:
The form of the equation is similar to the equation . Only here the parameters of the system are the length of the thread and the acceleration of gravity, and not the spring stiffness and the mass of the ball; the role of the coordinate is played by the length of the arc (i.e., the distance traveled, as in the first case).
Thus, free vibrations are described by equations of the same type (subject to the same laws) regardless of the physical nature of the forces causing these vibrations.
Solving equations and is a function of the form:
x = x mcos ω 0t(or x = x msin ω 0t).
That is, the coordinate of a body performing free oscillations changes over time according to the law of cosine or sine, and, therefore, these oscillations are harmonic:
In Eq. x = x mcos ω 0t(or x = x msin ω 0t), x m- vibration amplitude, ω 0 - own cyclic (circular) frequency of oscillations.
The cyclic frequency and period of free harmonic oscillations are determined by the properties of the system. Thus, for vibrations of a body attached to a spring, the following relations are valid:
.
The greater the spring stiffness or the smaller the mass of the load, the greater the natural frequency, which is fully confirmed by experience.
For a mathematical pendulum the following equalities are satisfied:
.
This formula was first obtained and tested experimentally by the Dutch scientist Huygens (a contemporary of Newton).
The period of oscillation increases with increasing length of the pendulum and does not depend on its mass.
Particular attention should be paid to the fact that harmonic oscillations are strictly periodic (since they obey the law of sine or cosine) and even for a mathematical pendulum, which is an idealization of a real (physical) pendulum, are possible only at small oscillation angles. If the deflection angles are large, the displacement of the load will not be proportional to the deflection angle (sine of the angle) and the acceleration will not be proportional to the displacement.
The speed and acceleration of a body oscillating freely will also undergo harmonic oscillations. Taking the time derivative of the function ( x = x mcos ω 0t(or x = x msin ω 0t)), we obtain an expression for speed:
v = -v msin ω 0t = -v mx mcos (ω 0t + π/2),
Where v m= ω 0 x m- velocity amplitude.
Similar expression for acceleration A we obtain by differentiating ( v = -v msin ω 0t = -v mx mcos (ω 0t + π/2)):
a = -a mcos ω 0t,
Where a m= ω 2 0x m- amplitude of acceleration. Thus, the amplitude of the speed of harmonic oscillations is proportional to the frequency, and the amplitude of acceleration is proportional to the square of the oscillation frequency.
| HARMONIC VIBRATIONS | ||
| Oscillations in which changes in physical quantities occur according to the law of cosine or sine (harmonic law), are called. harmonic vibrations. For example, in the case of mechanical harmonic vibrations:. In these formulas, ω is the frequency of vibration, x m is the amplitude of vibration, φ 0 and φ 0 ' are the initial phases of vibration. The above formulas differ in the definition of the initial phase and at φ 0 ’ = φ 0 +π/2 completely coincide. |   |
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| This is the simplest type of periodic oscillation. The specific form of the function (sine or cosine) depends on the method of removing the system from its equilibrium position. If the removal occurs by a push (kinetic energy is imparted), then at t=0 the displacement x=0, therefore, it is more convenient to use the sin function, setting φ 0 '=0; when deviating from the equilibrium position (potential energy is reported) at t = 0, the displacement x = x m, therefore, it is more convenient to use the cos function and φ 0 = 0. | ||
| The expression under the sign cos or sin is called. oscillation phase:. The phase of the oscillation is measured in radians and determines the value of the displacement (the oscillating quantity) at a given time. | ||
| The amplitude of the oscillation depends only on the initial deviation (the initial energy imparted to the oscillatory system). | ||
| Velocity and acceleration during harmonic oscillations. | ||
| According to the definition of speed, speed is the derivative of a position with respect to time | ||
| Thus, we see that the speed during harmonic oscillatory motion also changes according to the harmonic law, but the speed oscillations are ahead of the phase displacement oscillations by π/2. | ||
| Value - maximum speed of oscillatory motion (amplitude of speed fluctuations). | ||
Therefore, for the speed during harmonic oscillation we have:  , and for the case of zero initial phase (see graph). |  |
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According to the definition of acceleration, acceleration is the derivative of speed with respect to time:  is the second derivative of the coordinate with respect to time. Then: . Acceleration during harmonic oscillatory motion also changes according to the harmonic law, but acceleration oscillations are ahead of speed oscillations by π/2 and displacement oscillations by π (oscillations are said to occur in antiphase).
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Value - maximum acceleration (amplitude of acceleration fluctuations). Therefore, for acceleration we have:  , and for the case of zero initial phase:  (see chart). | ||
| From the analysis of the process of oscillatory motion, graphs and corresponding mathematical expressions, it is clear that when the oscillating body passes the equilibrium position (the displacement is zero), the acceleration is zero, and the speed of the body is maximum (the body passes the equilibrium position by inertia), and when the amplitude value of the displacement is reached, the speed is equal to zero, and the acceleration is maximum in absolute value (the body changes the direction of its movement). | ||
Let's compare the expressions for displacement and acceleration during harmonic vibrations: and  .
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You can write:  - i.e. the second derivative of the displacement is directly proportional (with the opposite sign) to the displacement. This equation is called equation of harmonic vibration. This dependence holds for any harmonic oscillation, regardless of its nature. Since we have never used the parameters of a specific oscillatory system, only the cyclic frequency can depend on them. | |
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It is often convenient to write the equations for vibrations in the form:  , where T is the oscillation period. Then, if time is expressed in fractions of a period, calculations will be simplified. For example, if we need to find the displacement after 1/8 of the period, we get: . Same for speed and acceleration. | |
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There are often cases when a system simultaneously participates in two or several oscillations independent of each other. In these cases, a complex oscillatory motion is formed, which is created by superimposing (adding) oscillations on each other. Obviously, cases of addition of oscillations can be very diverse. They depend not only on the number of added oscillations, but also on the parameters of the oscillations, on their frequencies, phases, amplitudes, and directions. It is not possible to review all the possible variety of cases of addition of oscillations, so we will limit ourselves to considering only individual examples.
1. Addition of oscillations of one direction. Let's add two oscillations of the same frequency, but different phases and amplitudes. 
(4.40)
When oscillations are superimposed on each other 
Let us introduce new parameters A and j according to the equations: 
(4.42)
System of equations (4.42) is easy to solve.
(4.43)
(4.44)
Thus, for x we finally obtain the equation
(4.45)
So, as a result of the addition of unidirectional oscillations of the same frequency, we obtain a harmonic (sinusoidal) oscillation, the amplitude and phase of which are determined by formulas (4.43) and (4.44).
Let us consider special cases in which the relationships between the phases of two added oscillations are different: 
(4.46)
Let us now add up unidirectional oscillations of the same amplitude, identical phases, but different frequencies. 
(4.47)
Let's consider the case when the frequencies are close to each other, i.e. w1~w2=w
Then we will approximately assume that (w1+w2)/2= w, and (w2-w1)/2 is a small value. The equation for the resulting oscillation will look like:
(4.48)
Its graph is shown in Fig. 4.5 This oscillation is called beating. It occurs with a frequency w, but its amplitude oscillates with a large period. 
2. Addition of two mutually perpendicular oscillations. Let us assume that one oscillation occurs along the x-axis, the other along the y-axis. The resulting motion is obviously located in the xy plane.
1. Let us assume that the oscillation frequencies and phases are the same, but the amplitudes are different. 
(4.49)
To find the trajectory of the resulting movement, you need to eliminate time from equations (4.49). To do this, it is enough to divide one equation term by term by another, as a result of which we get 
(4.50)
Equation (4.50) shows that in this case, the addition of oscillations leads to oscillation in a straight line, the slope of which is determined by the ratio of the amplitudes.
2. Let the phases of the added oscillations differ from each other by /2 and the equations have the form:
(4.51)
To find the trajectory of the resulting movement, excluding time, you need to square equations (4.51), first dividing them into A1 and A2, respectively, and then add them. The trajectory equation will take the form: 
(4.52)
This is the equation of an ellipse. It can be proven that for any initial phases and any amplitudes of two added mutually perpendicular oscillations of the same frequency, the resulting oscillation will occur along an ellipse. Its orientation will depend on the phases and amplitudes of the added oscillations.
If the added oscillations have different frequencies, then the trajectories of the resulting movements turn out to be very diverse. Only if the oscillation frequencies in x and y are multiples of each other, closed trajectories are obtained. Such movements can be classified as periodic. In this case, the trajectories of movements are called Lissajous figures. Let's consider one of the Lissajous figures, which is obtained by adding oscillations with frequency ratios of 1:2, with identical amplitudes and phases at the beginning of movement. 
(4.53)
Oscillations occur two times more often along the y-axis than along the x-axis. The addition of such oscillations will lead to a movement trajectory in the form of a figure eight (Fig. 4.7).
8. Damped oscillations and their parameters: decrement and oscillation coefficient, relaxation time
)Period of damped oscillations:
T = 
(58)
At δ << ω o vibrations do not differ from harmonic ones: T = 2π/ ω o.
2) Amplitude of damped oscillations is expressed by formula (119).
3) Decrement of attenuation, equal to the ratio of two successive vibration amplitudes A(t) And A(t+T), characterizes the rate of decrease in amplitude over a period:
= e d T (59)
4) Logarithmic damping decrement- natural logarithm of the ratio of the amplitudes of two successive oscillations corresponding to moments of time differing by a period
q = ln = ln e d Т =dT(60)
The logarithmic damping decrement is a constant value for a given oscillatory system.
5) Relaxation time it is customary to call the period of time ( t) during which the amplitude of damped oscillations decreases by e times:
e d τ = e, δτ = 1,
t = 1/d, (61)
From a comparison of expressions (60) and (61) we obtain:
q= = , (62)
Where N e - the number of oscillations performed during relaxation.
If during the time t the system commits Ν hesitation, then t = Ν . Τ and the equation of damped oscillations can be represented as:
S = A 0 e -d N T cos(w t+j)= A 0 e -q N cos(w t+j).
6)Quality factor of the oscillatory system(Q) is usually called the quantity characterizing the loss of energy in the system during the oscillation period:
Q = 2p , (63)
Where W- total energy of the system, ΔW- energy dissipated over a period. The less energy is dissipated, the greater the quality factor of the system. Calculations show that
Q = = pN e = = . (64)
However, the quality factor is inversely proportional to the logarithmic attenuation decrement. From formula (64) it follows that the quality factor is proportional to the number of oscillations N e performed by the system during relaxation.
7) Potential energy system at time t, can be expressed in terms of potential energy W 0 at greatest deviation:
W = = kA o 2 e -2 qN = W 0 e -2 qN . (65)
It is usually conventionally considered that the oscillations have practically stopped if their energy has decreased by 100 times (the amplitude has decreased by 10 times). From here we can obtain an expression for calculating the number of oscillations performed by the system:
= e 2qN= 100, ln100 = 2 qN;
N = = . (66)
9. Forced vibrations. Resonance. Aperiodic oscillations. Self-oscillations.
In order for the system to perform undamped oscillations, it is necessary to compensate for the loss of oscillation energy due to friction from the outside. In order to ensure that the oscillation energy of the system does not decrease, a force is usually introduced that periodically acts on the system (we will call such a force forcing, and the oscillations are forced).
DEFINITION: forced These are the oscillations that occur in an oscillatory system under the influence of an external periodically changing force.
This force usually plays a dual role:
firstly, it rocks the system and provides it with a certain amount of energy;
secondly, it periodically replenishes energy losses (energy consumption) to overcome the forces of resistance and friction.
Let the driving force change over time according to the law:
.
Let us compose an equation of motion for a system oscillating under the influence of such a force. We assume that the system is also affected by a quasi-elastic force and the resistance force of the medium (which is true under the assumption of small oscillations). Then the equation of motion of the system will look like:
Or 
.
Having made the substitutions , , – the natural frequency of oscillations of the system, we obtain an inhomogeneous linear differential equation 2 th order:
From the theory of differential equations it is known that the general solution of an inhomogeneous equation is equal to the sum of the general solution of a homogeneous equation and a particular solution of an inhomogeneous equation.
The general solution of the homogeneous equation is known:
,
Where 
; a 0 and a– arbitrary const.
.
Using a vector diagram, you can verify that this assumption is true, and also determine the values of “ a" And " j”.
The amplitude of oscillations is determined by the following expression:
.
Meaning " j”, which is the magnitude of the phase lag of the forced oscillation 
from the driving force that determined it, is also determined from the vector diagram and amounts to:
.
Finally, a particular solution to the inhomogeneous equation will take the form:
 | (8.18) |
This function, combined with
 | (8.19) |
gives a general solution to an inhomogeneous differential equation that describes the behavior of a system under forced oscillations. The term (8.19) plays a significant role in the initial stage of the process, during the so-called establishment of oscillations (Fig. 8.10). Over time, due to the exponential factor, the role of the second term (8.19) decreases more and more, and after sufficient time it can be neglected, retaining only the term (8.18) in the solution.
Thus, function (8.18) describes steady-state forced oscillations. They represent harmonic oscillations with a frequency equal to the frequency of the driving force. The amplitude of forced oscillations is proportional to the amplitude of the driving force. For a given oscillatory system (defined by w 0 and b), the amplitude depends on the frequency of the driving force. Forced oscillations lag behind the driving force in phase, and the magnitude of the lag “j” also depends on the frequency of the driving force.
The dependence of the amplitude of forced oscillations on the frequency of the driving force leads to the fact that at a certain frequency determined for a given system, the amplitude of oscillations reaches a maximum value. The oscillatory system turns out to be especially responsive to the action of the driving force at this frequency. This phenomenon is called resonance, and the corresponding frequency is resonant frequency.
DEFINITION: the phenomenon in which a sharp increase in the amplitude of forced oscillations is observed is called resonance.
The resonant frequency is determined from the maximum condition for the amplitude of forced oscillations:
. (8.20)
Then, substituting this value into the expression for the amplitude, we get:
. (8.21)
In the absence of medium resistance, the amplitude of oscillations at resonance would turn to infinity; the resonant frequency under the same conditions (b=0) coincides with the natural frequency of oscillations.
The dependence of the amplitude of forced oscillations on the frequency of the driving force (or, what is the same, on the oscillation frequency) can be represented graphically (Fig. 8.11). The individual curves correspond to different values of “b”. The smaller “b”, the higher and to the right the maximum of this curve lies (see the expression for w res.). With very high damping, resonance is not observed - with increasing frequency, the amplitude of forced oscillations monotonically decreases (lower curve in Fig. 8.11).
The set of presented graphs corresponding to different values of b is called resonance curves.
Notes regarding resonance curves:
as w®0 tends, all curves come to the same nonzero value, equal to . This value represents the displacement from the equilibrium position that the system receives under the influence of a constant force F 0 .
as w®¥ all curves asymptotically tend to zero, because at high frequencies, the force changes its direction so quickly that the system does not have time to noticeably shift from its equilibrium position.
the smaller b, the more the amplitude near resonance changes with frequency, the “sharper” the maximum.
The phenomenon of resonance often turns out to be useful, especially in acoustics and radio engineering.
Self-oscillations- undamped oscillations in a dissipative dynamic system with nonlinear feedback, supported by constant energy, that is non-periodic external influence.
Self-oscillations differ from forced oscillations because the latter are caused periodic external influence and occur with the frequency of this influence, while the occurrence of self-oscillations and their frequency are determined by the internal properties of the self-oscillating system itself.
Term self-oscillations introduced into Russian terminology by A. A. Andronov in 1928.
Examples[
Examples of self-oscillations include:
· undamped oscillations of the clock pendulum due to the constant action of the gravity of the winding weight;
violin string vibrations under the influence of a uniformly moving bow
· the occurrence of alternating current in multivibrator circuits and other electronic generators at a constant supply voltage;
· oscillation of the air column in the pipe of the organ, with a uniform supply of air into it. (see also Standing wave)
· rotational vibrations of a brass clock gear with a steel axis suspended from a magnet and twisted (Gamazkov’s experiment) (the kinetic energy of the wheel, as in a unipolar generator, is converted into the potential energy of an electric field, the potential energy of the electric field, as in a unipolar motor, is converted into the kinetic energy of the wheel etc.)
Maklakov's hammer
A hammer that strikes using alternating current energy with a frequency many times lower than the frequency of the current in an electrical circuit.
The coil L of the oscillating circuit is placed above the table (or other object that needs to be struck). An iron tube enters from below, the lower end of which is the striking part of the hammer. The tube has a vertical slot to reduce Foucault currents. The parameters of the oscillatory circuit are such that the natural frequency of its oscillations coincides with the frequency of the current in the circuit (for example, alternating city current, 50 hertz).
After turning on the current and establishing oscillations, a resonance of the currents of the circuit and the external circuit is observed, and the iron tube is drawn into the coil. The inductance of the coil increases, the oscillatory circuit goes out of resonance, and the amplitude of current oscillations in the coil decreases. Therefore, the tube returns to its original position - outside the coil - under the influence of gravity. Then the current oscillations inside the circuit begin to increase, and resonance occurs again: the tube is again drawn into the coil.
The tube makes self-oscillations, that is, periodic movements up and down, and at the same time loudly knocks on the table, like a hammer. The period of these mechanical self-oscillations is tens of times longer than the period of the alternating current that supports them.
The hammer is named after M.I. Maklakov, a lecture assistant at the Moscow Institute of Physics and Technology, who proposed and carried out such an experiment to demonstrate self-oscillations.
Self-oscillation mechanism
Fig 1. Self-oscillation mechanism
Self-oscillations can have a different nature: mechanical, thermal, electromagnetic, chemical. The mechanism for the occurrence and maintenance of self-oscillations in different systems can be based on different laws of physics or chemistry. For an accurate quantitative description of self-oscillations of different systems, different mathematical apparatus may be required. Nevertheless, it is possible to imagine a diagram common to all self-oscillating systems that qualitatively describes this mechanism (Fig. 1).
On the diagram: S- source of constant (non-periodic) impact; R- a nonlinear controller that converts a constant effect into a variable one (for example, into an intermittent one in time), which “swings” oscillator V- oscillating element(s) of the system, and oscillator oscillations through feedback B control the operation of the regulator R, asking phase And frequency his actions. Dissipation (energy dissipation) in a self-oscillating system is compensated by the flow of energy into it from a source of constant influence, due to which the self-oscillations do not die out.
Rice. 2 Diagram of the ratchet mechanism of a pendulum clock
If the oscillating element of the system is capable of its own damped oscillations(so-called harmonic dissipative oscillator), self-oscillations (with equal dissipation and energy input into the system during the period) are established at a frequency close to resonant for this oscillator, their shape becomes close to harmonic, and the amplitude, in a certain range of values, the greater the magnitude of the constant external influence.
An example of this kind of system is the ratchet mechanism of a pendulum clock, the diagram of which is shown in Fig. 2. On the ratchet wheel axle A(which in this system performs the function of a nonlinear regulator) there is a constant moment of force M, transmitted through a gear train from the mainspring or from a weight. When the wheel rotates A its teeth impart short-term impulses of force to the pendulum P(oscillator), thanks to which its oscillations do not fade. The kinematics of the mechanism plays the role of feedback in the system, synchronizing the rotation of the wheel with the oscillations of the pendulum in such a way that during the full period of oscillation the wheel rotates through an angle corresponding to one tooth.
Self-oscillating systems that do not contain harmonic oscillators are called relaxation. The vibrations in them can be very different from harmonic ones, and have a rectangular, triangular or trapezoidal shape. The amplitude and period of relaxation self-oscillations are determined by the ratio of the magnitude of the constant impact and the characteristics of inertia and dissipation of the system.
Rice. 3 Electric bell
The simplest example of relaxation self-oscillations is the operation of an electric bell, shown in Fig. 3. The source of constant (non-periodic) exposure here is an electric battery U; The role of a nonlinear regulator is performed by a chopper T, closing and opening an electrical circuit, as a result of which an intermittent current appears in it; oscillating elements are a magnetic field periodically induced in the core of an electromagnet E, and anchor A, moving under the influence of an alternating magnetic field. The oscillations of the armature activate the breaker, which forms feedback.
The inertia of this system is determined by two different physical quantities: the moment of inertia of the armature A and inductance of the electromagnet winding E. An increase in any of these parameters leads to an increase in the period of self-oscillations.
If there are several elements in the system that oscillate independently of each other and simultaneously influence a nonlinear regulator or regulators (of which there may also be several), self-oscillations can take on a more complex nature, for example, aperiodic, or dynamic chaos.
In nature and technology
Self-oscillations underlie many natural phenomena:
· vibrations of plant leaves under the influence of a uniform air flow;
· formation of turbulent flows on river rifts and rapids;
· action of regular geysers, etc.
The operating principle of a large number of various technical devices and devices is based on self-oscillations, including:
· operation of all kinds of clocks, both mechanical and electrical;
· the sound of all wind and stringed musical instruments;
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The simplest type of oscillations are harmonic vibrations- oscillations in which the displacement of the oscillating point from the equilibrium position changes over time according to the law of sine or cosine.
Thus, with a uniform rotation of the ball in a circle, its projection (shadow in parallel rays of light) performs a harmonic oscillatory motion on a vertical screen (Fig. 1).
The displacement from the equilibrium position during harmonic vibrations is described by an equation (it is called the kinematic law of harmonic motion) of the form:
where x is the displacement - a quantity characterizing the position of the oscillating point at time t relative to the equilibrium position and measured by the distance from the equilibrium position to the position of the point at a given time; A - amplitude of oscillations - maximum displacement of the body from the equilibrium position; T - period of oscillation - time of one complete oscillation; those. the shortest period of time after which the values of physical quantities characterizing the oscillation are repeated; - initial phase;
Oscillation phase at time t. The oscillation phase is an argument of a periodic function, which, for a given oscillation amplitude, determines the state of the oscillatory system (displacement, speed, acceleration) of the body at any time.
If at the initial moment of time the oscillating point is maximally displaced from the equilibrium position, then , and the displacement of the point from the equilibrium position changes according to the law
If the oscillating point at is in a position of stable equilibrium, then the displacement of the point from the equilibrium position changes according to the law
The value V, the inverse of the period and equal to the number of complete oscillations completed in 1 s, is called the oscillation frequency:
If during time t the body makes N complete oscillations, then
Size 
showing how many oscillations a body makes in s is called cyclic (circular) frequency.
The kinematic law of harmonic motion can be written as:
Graphically, the dependence of the displacement of an oscillating point on time is represented by a cosine wave (or sine wave).
Figure 2, a shows a graph of the time dependence of the displacement of the oscillating point from the equilibrium position for the case.
Let's find out how the speed of an oscillating point changes with time. To do this, we find the time derivative of this expression:
where is the amplitude of the velocity projection onto the x-axis.
This formula shows that during harmonic oscillations, the projection of the body’s velocity onto the x-axis also changes according to a harmonic law with the same frequency, with a different amplitude and is ahead of the displacement in phase by (Fig. 2, b).
To clarify the dependence of acceleration, we find the time derivative of the velocity projection:
where is the amplitude of the acceleration projection onto the x-axis.
With harmonic oscillations, the acceleration projection is ahead of the phase displacement by k (Fig. 2, c).
Similarly, you can build dependency graphs
Considering that , the formula for acceleration can be written
those. with harmonic oscillations, the projection of acceleration is directly proportional to the displacement and is opposite in sign, i.e. acceleration is directed in the direction opposite to the displacement.
So, the acceleration projection is the second derivative of the displacement, then the resulting relationship can be written as:
The last equality is called harmonic equation.
A physical system in which harmonic oscillations can exist is called harmonic oscillator, and the equation of harmonic vibrations is harmonic oscillator equation.
HARMONIC VIBRATIONAL MOTION
§1 Kinematics of harmonic oscillation
Processes that repeat over time are called oscillations.
Depending on the nature of the oscillatory process and the excitation mechanism, there are: mechanical vibrations (oscillations of pendulums, strings, buildings, the earth's surface, etc.); electromagnetic oscillations (alternating current oscillations, oscillations of vectors and in an electromagnetic wave, etc.); electromechanical vibrations (vibrations of the telephone membrane, loudspeaker diffuser, etc.); vibrations of nuclei and molecules as a result of thermal motion in atoms.
Let's consider the segment [OD] (radius vector) performing rotational motion around point 0. Length |OD| = A . Rotation occurs with a constant angular velocity ω 0. Then the angle φ between the radius vector and the axisxchanges over time according to law
where φ 0 - angle between [OD] and axis X at a point in timet= 0. Projection of the segment [OD] onto the axis X at a point in timet= 0
and at an arbitrary moment in time
(1)
Thus, the projection of the segment [OD] onto the x axis undergoes oscillations occurring along the axis X, and these oscillations are described by the cosine law (formula (1)).
Oscillations that are described by the law of cosine
or sine
called harmonic.
Harmonic vibrations are periodic, because the value of x (and y) is repeated at regular intervals.
If the segment [OD] is in the lowest position in the figure, i.e. dot D coincides with the point R, then its projection onto the x axis is zero. Let us call this position of the segment [OD] the equilibrium position. Then we can say that the quantity X describes the displacement of an oscillating point from its equilibrium position. The maximum displacement from the equilibrium position is called amplitude fluctuations
Magnitude
which is under the cosine sign is called phase. Phase determines the displacement from the equilibrium position at an arbitrary moment in timet. Phase at the initial moment of timet = 0 , equal to φ 0 is called the initial phase.
T
The period of time during which one complete oscillation occurs is called the period of oscillation T. The number of oscillations per unit time is called the oscillation frequency ν.
After a period of time equal to the period T, i.e. when the cosine argument increases by ω 0 T, the movement is repeated, and the cosine takes on its previous value
because the period of the cosine is 2π, then, therefore, ω 0 T= 2π
thus, ω 0 is the number of oscillations of the body in 2π seconds. ω 0 - cyclic or circular frequency.
harmonic vibration pattern
A- amplitude, T- period, X- displacement,t- time.
We find the speed of the oscillating point by differentiating the displacement equation X(t) by time
those. speed vdifferent in phase from the offset X onπ /2.
Acceleration is the first derivative of velocity (second derivative of displacement) with respect to time
those. acceleration A differs from the phase shift by π.
Let's build a graph X(
t)
, y(
t)
And A(
t)
in one coordinate estimate (for simplicity, let’s take φ 0 = 0 and ω 0 = 1)
Free or own are called oscillations that occur in a system left to itself after it has been removed from its equilibrium position.
Any periodically repeating movement is called oscillatory. Therefore, the dependences of the coordinates and speed of a body on time during oscillations are described by periodic functions of time. In the school physics course, vibrations are considered in which the dependencies and velocities of the body are trigonometric functions 
, 
or a combination thereof, where is a certain number. Such oscillations are called harmonic (functions 
And 
often called harmonic functions). To solve problems on oscillations included in the program of the unified state exam in physics, you need to know the definitions of the main characteristics of oscillatory motion: amplitude, period, frequency, circular (or cyclic) frequency and phase of oscillations. Let us give these definitions and connect the listed quantities with the parameters of the dependence of the body coordinates on time, which in the case of harmonic oscillations can always be represented in the form
where , and are some numbers.
The amplitude of oscillations is the maximum deviation of an oscillating body from its equilibrium position. Since the maximum and minimum values of the cosine in (11.1) are equal to ±1, the amplitude of oscillations of the body oscillating (11.1) is equal to . The period of oscillation is the minimum time after which the movement of a body is repeated. For dependence (11.1), the period can be set from the following considerations. Cosine is a periodic function with period. Therefore, the movement is completely repeated through such a value that . From here we get
The circular (or cyclic) frequency of oscillations is the number of oscillations performed per unit of time. From formula (11.3) we conclude that the circular frequency is the quantity from formula (11.1).
The oscillation phase is the argument of a trigonometric function that describes the dependence of the coordinate on time. From formula (11.1) we see that the phase of oscillations of the body, the movement of which is described by dependence (11.1), is equal to 
. The value of the oscillation phase at time = 0 is called the initial phase. For dependence (11.1), the initial phase of oscillations is equal to . Obviously, the initial phase of oscillations depends on the choice of the time reference point (moment = 0), which is always conditional. By changing the origin of time, the initial phase of oscillations can always be “made” equal to zero, and the sine in formula (11.1) can be “turned” into a cosine or vice versa.
The Unified State Exam program also includes knowledge of formulas for the frequency of oscillations of spring and mathematical pendulums. A spring pendulum is usually called a body that can oscillate on a smooth horizontal surface under the action of a spring, the second end of which is fixed (left figure). A mathematical pendulum is a massive body, the dimensions of which can be neglected, oscillating on a long, weightless and inextensible thread (right figure). The name of this system, “mathematical pendulum,” is due to the fact that it represents an abstract mathematical model of real ( physical) pendulum. It is necessary to remember the formulas for the period (or frequency) of oscillations of spring and mathematical pendulums. For a spring pendulum
where is the length of the thread, is the acceleration of gravity. Let's consider the application of these definitions and laws using the example of problem solving.
To find the cyclic frequency of oscillations of the load in task 11.1.1 Let's first find the period of oscillation, and then use formula (11.2). Since 10 m 28 s is 628 s, and during this time the load oscillates 100 times, the period of oscillation of the load is 6.28 s. Therefore, the cyclic frequency of oscillations is 1 s -1 (answer 2 ). IN problem 11.1.2 the load made 60 oscillations in 600 s, so the oscillation frequency is 0.1 s -1 (answer 1 ).
To understand the distance the load will travel in 2.5 periods ( problem 11.1.3), let's follow his movement. After a period, the load will return back to the point of maximum deflection, completing a complete oscillation. Therefore, during this time, the load will travel a distance equal to four amplitudes: to the equilibrium position - one amplitude, from the equilibrium position to the point of maximum deviation in the other direction - the second, back to the equilibrium position - the third, from the equilibrium position to the starting point - the fourth. During the second period, the load will again go through four amplitudes, and during the remaining half of the period - two amplitudes. Therefore, the distance traveled is equal to ten amplitudes (answer 4 ).
The amount of movement of the body is the distance from the starting point to the ending point. Over 2.5 periods in task 11.1.4 the body will have time to complete two full and half a full oscillation, i.e. will be at the maximum deviation, but on the other side of the equilibrium position. Therefore, the magnitude of the displacement is equal to two amplitudes (answer 3 ).
By definition, the oscillation phase is the argument of a trigonometric function that describes the dependence of the coordinates of an oscillating body on time. Therefore the correct answer is problem 11.1.5 - 3 .
A period is the time of complete oscillation. This means that the return of a body back to the same point from which the body began to move does not mean that a period has passed: the body must return to the same point with the same speed. For example, a body, having started oscillations from an equilibrium position, will have time to deviate by a maximum amount in one direction, return back, deviate by a maximum in the other direction, and return back again. Therefore, during the period the body will have time to deviate by the maximum amount from the equilibrium position twice and return back. Consequently, the passage from the equilibrium position to the point of maximum deviation ( problem 11.1.6) the body spends a quarter of the period (answer 3 ).
Harmonic oscillations are those in which the dependence of the coordinates of the oscillating body on time is described by a trigonometric (sine or cosine) function of time. IN task 11.1.7 these are the functions and , despite the fact that the parameters included in them are designated as 2 and 2 . The function is a trigonometric function of the square of time. Therefore, vibrations of only quantities and are harmonic (answer 4 ).
During harmonic vibrations, the speed of the body changes according to the law 
, where is the amplitude of the speed oscillations (the time reference point is chosen so that the initial phase of the oscillations is equal to zero). From here we find the dependence of the kinetic energy of the body on time 
(problem 11.1.8). Using further the well-known trigonometric formula, we obtain
From this formula it follows that the kinetic energy of a body changes during harmonic oscillations also according to the harmonic law, but with double the frequency (answer 2 ).
Behind the relationship between the kinetic energy of the load and the potential energy of the spring ( problem 11.1.9) is easy to follow from the following considerations. When the body is deflected by the maximum amount from the equilibrium position, the speed of the body is zero, and, therefore, the potential energy of the spring is greater than the kinetic energy of the load. On the contrary, when the body passes through the equilibrium position, the potential energy of the spring is zero, and therefore the kinetic energy is greater than the potential energy. Therefore, between the passage of the equilibrium position and the maximum deflection, the kinetic and potential energy are compared once. And since during a period the body passes four times from the equilibrium position to the maximum deflection or back, then during the period the kinetic energy of the load and the potential energy of the spring are compared with each other four times (answer 2 ).
Amplitude of speed fluctuations ( task 11.1.10) is easiest to find using the law of conservation of energy. At the point of maximum deflection, the energy of the oscillatory system is equal to the potential energy of the spring 
, where is the spring stiffness coefficient, is the vibration amplitude. When passing through the equilibrium position, the energy of the body is equal to the kinetic energy 
, where is the mass of the body, is the speed of the body when passing through the equilibrium position, which is the maximum speed of the body during the oscillation process and, therefore, represents the amplitude of the speed oscillations. Equating these energies, we find
(answer 4 ).
From formula (11.5) we conclude ( problem 11.2.2), that its period does not depend on the mass of a mathematical pendulum, and with an increase in length by 4 times, the period of oscillations increases by 2 times (answer 1 ).
A clock is an oscillatory process that is used to measure intervals of time ( problem 11.2.3). The words “clock is in a hurry” mean that the period of this process is less than what it should be. Therefore, to clarify the progress of these clocks, it is necessary to increase the period of the process. According to formula (11.5), to increase the period of oscillation of a mathematical pendulum, it is necessary to increase its length (answer 3 ).
To find the amplitude of oscillations in problem 11.2.4, it is necessary to represent the dependence of the body coordinates on time in the form of a single trigonometric function. For the function given in the condition, this can be done by introducing an additional angle. Multiplying and dividing this function by 
and using the formula for adding trigonometric functions, we get
|
where is the angle such that 
. From this formula it follows that the amplitude of body oscillations is 
(answer 4
).
