Varies over time according to a sinusoidal law:
Where X- the value of the fluctuating quantity at the moment of time t, A- amplitude, ω - circular frequency, φ — initial phase of oscillations, ( φt + φ ) - full phase of oscillations. At the same time, the values A, ω And φ - permanent.
For mechanical vibrations of fluctuating magnitude X are, in particular, displacement and speed, for electrical vibrations - voltage and current.
Harmonic vibrations occupy special place among all types of oscillations, since this is the only type of oscillations whose shape is not distorted when passing through any homogeneous medium, i.e., waves propagating from the source of harmonic oscillations will also be harmonic. Any non-harmonic oscillation can be represented as a sum (integral) of various harmonic oscillations (in the form of a spectrum of harmonic oscillations).
Energy transformations during harmonic vibrations.
During the oscillation process, potential energy transfer occurs Wp to kinetic Wk and vice versa. At the position of maximum deviation from the equilibrium position, the potential energy is maximum, the kinetic energy is zero. As it returns to the equilibrium position, the speed of the oscillating body increases, and with it the kinetic energy also increases, reaching a maximum in the equilibrium position. The potential energy drops to zero. Further movement occurs with a decrease in speed, which drops to zero when the deflection reaches its second maximum. The potential energy here increases to its initial (maximum) value (in the absence of friction). Thus, the oscillations of kinetic and potential energies occur with double the frequency (compared to the oscillations of the pendulum itself) and are in antiphase (i.e., there is a phase shift between them equal to π ). Total vibration energy W remains unchanged. For a body oscillating under the action of an elastic force, it is equal to:
Where v m— maximum body speed (in the equilibrium position), x m = A- amplitude.
Due to the presence of friction and resistance of the medium, free vibrations attenuate: their energy and amplitude decrease over time. Therefore, in practice, they often use not free, but forced oscillations.
Changes in any quantity are described using the laws of sine or cosine, then such oscillations are called harmonic. Let's consider a circuit consisting of a capacitor (which was charged before being included in the circuit) and an inductor (Fig. 1).
Picture 1.
The harmonic vibration equation can be written as follows:
$q=q_0cos((\omega )_0t+(\alpha )_0)$ (1)
where $t$ is time; $q$ charge, $q_0$-- maximum deviation of charge from its average (zero) value during changes; $(\omega )_0t+(\alpha )_0$- oscillation phase; $(\alpha )_0$- initial phase; $(\omega )_0$ - cyclic frequency. During the period, the phase changes by $2\pi $.
Equation of the form:
equation of harmonic oscillations in differential form for an oscillatory circuit that will not contain active resistance.
Any kind periodic oscillations can be accurately represented as the sum of harmonic vibrations, the so-called harmonic series.
For the oscillation period of a circuit that consists of a coil and a capacitor, we obtain Thomson’s formula:
If we differentiate expression (1) with respect to time, we can obtain the formula for the function $I(t)$:
The voltage across the capacitor can be found as:
From formulas (5) and (6) it follows that the current strength is ahead of the voltage on the capacitor by $\frac(\pi )(2).$
Harmonic oscillations can be represented both in the form of equations, functions and vector diagrams.
Equation (1) represents free undamped oscillations.
Damped Oscillation Equation
The change in charge ($q$) on the capacitor plates in the circuit, taking into account the resistance (Fig. 2), will be described by a differential equation of the form:
Figure 2.
If the resistance that is part of the circuit $R\
where $\omega =\sqrt(\frac(1)(LC)-\frac(R^2)(4L^2))$ is the cyclic oscillation frequency. $\beta =\frac(R)(2L)-$damping coefficient. The amplitude of damped oscillations is expressed as:
If at $t=0$ the charge on the capacitor is equal to $q=q_0$ and there is no current in the circuit, then for $A_0$ we can write:
The phase of oscillations at the initial moment of time ($(\alpha )_0$) is equal to:
When $R >2\sqrt(\frac(L)(C))$ the change in charge is not an oscillation, the discharge of the capacitor is called aperiodic.
Example 1
Exercise: The maximum charge value is $q_0=10\ C$. It varies harmonically with a period of $T= 5 s$. Determine the maximum possible current.
Solution:
As a basis for solving the problem we use:
To find the current strength, expression (1.1) must be differentiated with respect to time:
where the maximum (amplitude value) of the current strength is the expression:
From the conditions of the problem we know the amplitude value of the charge ($q_0=10\ C$). You should find the natural frequency of oscillations. Let's express it as:
\[(\omega )_0=\frac(2\pi )(T)\left(1.4\right).\]
In this case, the desired value will be found using equations (1.3) and (1.2) as:
Since all quantities in the problem conditions are presented in the SI system, we will carry out the calculations:
Answer:$I_0=12.56\ A.$
Example 2
Exercise: What is the period of oscillation in a circuit that contains an inductor $L=1$H and a capacitor, if the current strength in the circuit changes according to the law: $I\left(t\right)=-0.1sin20\pi t\ \left(A \right)?$ What is the capacitance of the capacitor?
Solution:
From the equation of current fluctuations, which is given in the conditions of the problem:
we see that $(\omega )_0=20\pi $, therefore, we can calculate the Oscillation period using the formula:
\ \
According to Thomson's formula for a circuit that contains an inductor and a capacitor, we have:
Let's calculate the capacity:
Answer:$T=0.1$ c, $C=2.5\cdot (10)^(-4)F.$
Oscillations are a process of changing the states of a system around the equilibrium point that is repeated to varying degrees over time.
Harmonic oscillation - oscillations in which a physical (or any other) quantity changes over time according to a sinusoidal or cosine law. The kinematic equation of harmonic oscillations has the form
where x is the displacement (deviation) of the oscillating point from the equilibrium position at time t; A is the amplitude of oscillations, this is the value that determines the maximum deviation of the oscillating point from the equilibrium position; ω - cyclic frequency, a value indicating the number of complete oscillations occurring within 2π seconds - the full phase of oscillations, 0 - the initial phase of oscillations.
Amplitude is the maximum value of displacement or change of a variable from the average value during oscillatory or wave motion.
The amplitude and initial phase of oscillations are determined by the initial conditions of movement, i.e. position and speed of the material point at the moment t=0.
Generalized harmonic oscillation in differential form
the amplitude of sound waves and audio signals usually refers to the amplitude of the air pressure in the wave, but is sometimes described as the amplitude of the displacement relative to equilibrium (the air or the speaker's diaphragm)
Frequency is a physical quantity, a characteristic of a periodic process, equal to the number of complete cycles of the process completed per unit of time. The frequency of vibration in sound waves is determined by the frequency of vibration of the source. High frequency oscillations decay faster than low frequency ones.
The reciprocal of the oscillation frequency is called period T.
The period of oscillation is the duration of one complete cycle of oscillation.
In the coordinate system, from point 0 we draw a vector A̅, the projection of which onto the OX axis is equal to Аcosϕ. If the vector A̅ rotates uniformly with an angular velocity ω˳ counterclockwise, then ϕ=ω˳t +ϕ˳, where ϕ˳ is the initial value of ϕ (oscillation phase), then the amplitude of the oscillations is the modulus of the uniformly rotating vector A̅, the oscillation phase (ϕ ) - angle between vector A̅ and OX axis, initial phase (ϕ˳) -initial value of this angle, the angular frequency of oscillations (ω) is the angular velocity of rotation of the vector A̅..
2. Characteristics of wave processes: wave front, beam, wave speed, wave length. Longitudinal and transverse waves; examples.
The surface dividing this moment time, the medium already covered and not yet covered by oscillations is called the wave front. At all points of such a surface, after the wave front leaves, oscillations are established that are identical in phase.
The beam is perpendicular to the wave front. Acoustic rays, like light rays, are rectilinear in a homogeneous medium. They are reflected and refracted at the interface between 2 media.
Wavelength is the distance between two points closest to each other, oscillating in the same phases, usually the wavelength is denoted by the Greek letter. By analogy with waves created in water by a thrown stone, the wavelength is the distance between two adjacent wave crests. One of the main characteristics of vibrations. Measured in distance units (meters, centimeters, etc.)
- longitudinal waves (compression waves, P-waves) - particles of the medium vibrate parallel(along) the direction of wave propagation (as, for example, in the case of sound propagation);
- transverse waves (shear waves, S-waves) - particles of the medium vibrate perpendicular direction of wave propagation ( electromagnetic waves, waves on separation surfaces);
The angular frequency of oscillations (ω) is the angular velocity of rotation of the vector A̅(V), the displacement x of the oscillating point is the projection of the vector A onto the OX axis.
V=dx/dt=-Aω˳sin(ω˳t+ϕ˳)=-Vmsin(ω˳t+ϕ˳), where Vm=Аω˳ is the maximum speed (velocity amplitude)
3. Free and forced vibrations. Natural frequency of oscillations of the system. The phenomenon of resonance. Examples .
Free (natural) vibrations are called those that occur without external influences due to the energy initially obtained by heat. Characteristic models of such mechanical oscillations are a material point on a spring (spring pendulum) and a material point on an inextensible thread (mathematical pendulum).
In these examples, oscillations arise either due to initial energy (deviation of a material point from the position of equilibrium and motion without initial speed), or due to kinetic (the body is imparted speed in the initial equilibrium position), or due to both energy (imparting speed to the body deviated from the equilibrium position).
Consider a spring pendulum. In the equilibrium position, the elastic force F1
balances the force of gravity mg. If you pull the spring a distance x, then a large elastic force will act on the material point. The change in the value of the elastic force (F), according to Hooke's law, is proportional to the change in the length of the spring or the displacement x of the point: F= - rx
Another example. The mathematical pendulum of deviation from the equilibrium position is such a small angle α that the trajectory of a material point can be considered a straight line coinciding with the OX axis. In this case, the approximate equality is satisfied: α ≈sin α≈ tanα ≈x/L
Undamped oscillations. Let us consider a model in which the resistance force is neglected.
The amplitude and initial phase of oscillations are determined by the initial conditions of movement, i.e. position and speed of the material point moment t=0.
Among various types Harmonic vibration is the simplest form of vibration.
Thus, a material point suspended on a spring or thread performs harmonic oscillations, if resistance forces are not taken into account.
The period of oscillation can be found from the formula: T=1/v=2П/ω0
Damped oscillations. In a real case, resistance (friction) forces act on an oscillating body, the nature of the movement changes, and the oscillation becomes damped.
In relation to one-dimensional motion, we give the last formula the following form: Fc = - r * dx/dt
The rate at which the oscillation amplitude decreases is determined by the damping coefficient: the stronger the braking effect of the medium, the greater ß and the faster the amplitude decreases. In practice, however, the degree of damping is often characterized by a logarithmic damping decrement, meaning by this a value equal to the natural logarithm of the ratio of two successive amplitudes separated by a time interval equal to the oscillation period; therefore, the damping coefficient and the logarithmic damping decrement are related by a fairly simple relationship: λ=ßT
With strong damping, it is clear from the formula that the period of oscillation is an imaginary quantity. The movement in this case will no longer be periodic and is called aperiodic.
Forced vibrations. Forced oscillations are called oscillations that occur in a system with the participation of an external force that changes according to a periodic law.
Let us assume that the material point, in addition to the elastic force and the friction force, is acted upon by an external driving force F=F0 cos ωt
The amplitude of the forced oscillation is directly proportional to the amplitude of the driving force and has a complex dependence on the damping coefficient of the medium and the circular frequencies of natural and forced oscillations. If ω0 and ß are given for the system, then the amplitude of forced oscillations has a maximum value at some specific frequency of the driving force, called resonant The phenomenon itself—the achievement of the maximum amplitude of forced oscillations for given ω0 and ß—is called resonance.
The resonant circular frequency can be found from the condition of the minimum denominator in: ωres=√ωₒ- 2ß
Mechanical resonance can be both beneficial and harmful. The harmful effects are mainly due to the destruction it can cause. Thus, in technology, taking into account various vibrations, it is necessary to provide for the possible occurrence of resonant conditions, otherwise there may be destruction and disasters. Bodies usually have several natural vibration frequencies and, accordingly, several resonant frequencies.
Resonance phenomena under the action of external mechanical vibrations occur in internal organs. This is apparently one of the reasons for the negative impact of infrasonic vibrations and vibrations on the human body.
6.Sound research methods in medicine: percussion, auscultation. Phonocardiography.
Sound can be a source of status information internal organs human, therefore in medicine such methods of studying the patient’s condition as auscultation, percussion and phonocardiography are widely used
Auscultation
For auscultation, a stethoscope or phonendoscope is used. A phonendoscope consists of a hollow capsule with a sound-transmitting membrane that is applied to the patient’s body, from which rubber tubes go to the doctor’s ear. A resonance of the air column occurs in the capsule, resulting in increased sound and improved auscultation. When auscultating the lungs, breathing sounds and various wheezing characteristic of diseases are heard. You can also listen to the heart, intestines and stomach.
Percussion
In this method, the sound of individual parts of the body is listened to by tapping them. Let's imagine a closed cavity inside some body, filled with air. If you induce sound vibrations in this body, then at a certain frequency of sound, the air in the cavity will begin to resonate, releasing and amplifying a tone corresponding to the size and position of the cavity. The human body can be represented as a collection of gas-filled (lungs), liquid (internal organs) and solid (bones) volumes. When hitting the surface of a body, vibrations occur, the frequencies of which have a wide range. From this range, some vibrations will fade out quite quickly, while others, coinciding with the natural vibrations of the voids, will intensify and, due to resonance, will be audible.
Phonocardiography
Used to diagnose cardiac conditions. The method consists of graphically recording heart sounds and murmurs and their diagnostic interpretation. A phonocardiograph consists of a microphone, an amplifier, a system of frequency filters and a recording device.
9. Ultrasound research methods (ultrasound) in medical diagnostics.
1) Diagnostic and research methods
These include location methods using mainly pulsed radiation. This is echoencephalography - detection of tumors and edema of the brain. Ultrasound cardiography – measurement of heart size in dynamics; in ophthalmology - ultrasonic location to determine the size of the ocular media.
2)Methods of influence
Ultrasound physiotherapy – mechanical and thermal effects on tissue.
11. Shock wave. Production and use of shock waves in medicine.
Shock wave
– a discontinuity surface that moves relative to the gas and upon crossing which the pressure, density, temperature and speed experience a jump.
Under large disturbances (explosion, supersonic movement of bodies, powerful electric discharge, etc.), the speed of oscillating particles of the medium can become comparable to the speed of sound , a shock wave occurs.
The shock wave can have significant energy, yes, at nuclear explosion for the formation of a shock wave in environment about 50% of the explosion energy is consumed. Therefore, a shock wave, reaching biological and technical objects, can cause death, injury and destruction.
Shock waves are used in medical technology, representing an extremely short, powerful pressure pulse with high pressure amplitudes and a small stretch component. They are generated outside the patient’s body and transmitted deep into the body, producing a therapeutic effect provided for by the specialization of the equipment model: crushing urinary stones, treating pain areas and the consequences of injuries to the musculoskeletal system, stimulating the recovery of the heart muscle after myocardial infarction, smoothing cellulite formations, etc.
>>Harmonic vibrations
§ 22 HARMONIC VIBRATIONS
Knowing how the acceleration and coordinate of an oscillating body are related to each other, it is possible, based on mathematical analysis, to find the dependence of the coordinate on time.
Acceleration is the second derivative of a coordinate with respect to time. Instantaneous speed a point, as you know from a mathematics course, is the derivative of the coordinates of a point with respect to time. The acceleration of a point is the derivative of its speed with respect to time, or the second derivative of the coordinate with respect to time. Therefore, equation (3.4) can be written as follows:
where x " - second derivative of the coordinate with respect to time. According to equation (3.11), during free oscillations, the coordinate x changes with time so that the second derivative of the coordinate with respect to time is directly proportional to the coordinate itself and is opposite in sign.
From the course of mathematics it is known that the second derivatives of sine and cosine with respect to their argument are proportional to the functions themselves, taken with the opposite sign. IN mathematical analysis it is proved that no other functions have this property. All this allows us to legitimately assert that the coordinate of a body performing free oscillations changes over time according to the law of sine or pasine. Figure 3.6 shows the change in the coordinate of a point over time according to the cosine law.
Periodic changes in a physical quantity depending on time, occurring according to the law of sine or cosine, are called harmonic oscillations.
Amplitude of oscillations. The amplitude of harmonic oscillations is the modulus of the greatest displacement of a body from its equilibrium position.
The amplitude may have different meanings depending on how much we displace the body from the equilibrium position at the initial moment of time, or on what speed is imparted to the body. The amplitude is determined by the initial conditions, or more precisely by the energy imparted to the body. But the maximum values of the sine modulus and cosine modulus are equal to one. Therefore, the solution to equation (3.11) cannot be expressed simply as a sine or cosine. It should take the form of the product of the oscillation amplitude x m by sine or cosine.
Solution of the equation describing free vibrations. Let us write the solution to equation (3.11) in the following form:
and the second derivative will be equal to:
We have obtained equation (3.11). Consequently, function (3.12) is a solution to the original equation (3.11). The solution to this equation will also be the function
The graph of the body coordinate versus time according to (3.14) is a cosine wave (see Fig. 3.6).
Period and frequency of harmonic oscillations. When oscillating, the body's movements are periodically repeated. The time period T during which the system completes one complete cycle of oscillations is called the period of oscillations.
Knowing the period, you can determine the frequency of oscillations, i.e. the number of oscillations per unit of time, for example per second. If one oscillation occurs in time T, then the number of oscillations per second
IN International system units (SI) the oscillation frequency is equal to one if one oscillation occurs per second. The unit of frequency is called the hertz (abbreviated: Hz) in honor of the German physicist G. Hertz.
The number of oscillations in 2 s is equal to:
The quantity is the cyclic, or circular, frequency of oscillations. If in equation (3.14) time t is equal to one period, then T = 2. Thus, if at time t = 0 x = x m, then at time t = T x = x m, i.e. through a period of time equal to one period, the oscillations are repeated.
The frequency of free vibrations is determined by the natural frequency of the oscillatory system 1.
Dependence of the frequency and period of free oscillations on the properties of the system. The natural frequency of vibration of a body attached to a spring, according to equation (3.13), is equal to:
The greater the spring stiffness k, the greater it is, and the less, the greater the body mass m. This is easy to understand: a stiff spring imparts greater acceleration to the body and changes the speed of the body faster. And the more massive the body, the slower it changes speed under the influence of force. The oscillation period is equal to:
Having a set of springs of different stiffness and bodies of different masses, it is easy to verify from experience that formulas (3.13) and (3.18) correctly describe the nature of the dependence of and T on k and m.
It is remarkable that the period of oscillation of a body on a spring and the period of oscillation of a pendulum at small angles of deflection do not depend on the amplitude of oscillations.
The modulus of the proportionality coefficient between the acceleration t and the displacement x in equation (3.10), which describes the oscillations of the pendulum, is, as in equation (3.11), the square of the cyclic frequency. Therefore, the natural frequency of oscillations mathematical pendulum at small angles of deviation of the thread from the vertical depends on the length of the pendulum and the acceleration of gravity:
This formula was first obtained and tested experimentally by the Dutch scientist G. Huygens, a contemporary of I. Newton. It is valid only for small angles of thread deflection.
1 Often in the following, for brevity, we will simply refer to the cyclic frequency as the frequency. You can distinguish the cyclic frequency from the normal frequency by notation.
The period of oscillation increases with increasing length of the pendulum. It does not depend on the mass of the pendulum. This can be easily verified experimentally with various pendulums. The dependence of the oscillation period on the acceleration of gravity can also be detected. The smaller g, the longer the period of oscillation of the pendulum and, therefore, the slower the pendulum clock runs. Thus, a clock with a pendulum in the form of a weight on a rod will fall behind by almost 3 s per day if it is lifted from the basement to the top floor of Moscow University (height 200 m). And this is only due to the decrease in the acceleration of free fall with height.
The dependence of the period of oscillation of a pendulum on the value of g is used in practice. By measuring the oscillation period, g can be determined very accurately. The acceleration of gravity changes with geographical latitude. But even at a given latitude it is not the same everywhere. After all, the density of the earth's crust is not the same everywhere. In areas where dense rocks occur, the acceleration g is somewhat greater. This is taken into account when searching for minerals.
Thus, iron ore has a higher density compared to ordinary rocks. Measurements of the acceleration of free fall near Kursk, carried out under the leadership of Academician A. A. Mikhailov, made it possible to clarify the location of the iron ore. They were first discovered through magnetic measurements.
The properties of mechanical vibrations are used in the devices of most electronic scales. The body to be weighed is placed on a platform under which a rigid spring is installed. As a result, mechanical vibrations arise, the frequency of which is measured by a corresponding sensor. The microprocessor associated with this sensor converts the oscillation frequency into the mass of the body being weighed, since this frequency depends on the mass.
The resulting formulas (3.18) and (3.20) for the oscillation period indicate that the period of harmonic oscillations depends on the system parameters (spring stiffness, thread length, etc.)
Myakishev G. Ya., Physics. 11th grade: educational. for general education institutions: basic and profile. levels / G. Ya. Myakishev, B. V. Bukhovtsev, V. M. Charugin; edited by V. I. Nikolaeva, N. A. Parfentieva. - 17th ed., revised. and additional - M.: Education, 2008. - 399 p.: ill.
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Definition of harmonic vibrations. Characteristics of harmonic oscillations: displacement from the equilibrium position, amplitude of oscillations, phase of oscillation, frequency and period of oscillations. Velocity and acceleration of an oscillating point. Energy of a harmonic oscillator. Examples of harmonic oscillators: mathematical, spring, torsional and physical Chinese pendulums.
Acoustics, radio engineering, optics and other branches of science and technology are based on the study of oscillations and waves. The theory of vibrations plays an important role in mechanics, especially in strength calculations of aircraft, bridges, and certain types of machines and components.
Oscillations are processes that repeat at regular intervals (and not all repeating processes are oscillations!). Depending on the physical nature of the repeating process, vibrations are distinguished between mechanical, electromagnetic, electromechanical, etc. During mechanical vibrations, the positions and coordinates of bodies periodically change.
restoring force - the force under the influence of which the oscillatory process occurs. This force tends to return a body or a material point, deviated from its position of rest, to its original position.
Depending on the nature of the impact on the oscillating body, a distinction is made between free (or natural) vibrations and forced vibrations.
Depending on the nature of the impact on the oscillating system, free oscillations, forced oscillations, self-oscillations and parametric oscillations are distinguished.
Free (own) oscillations are those oscillations that occur in a system left to itself after it has been given a push, or it has been removed from an equilibrium position, i.e. when only a restoring force acts on an oscillating body. An example is the oscillation of a ball suspended on a thread. In order to cause vibrations, you must either push the ball or, moving it to the side, release it. In the event that no energy dissipation occurs, free oscillations are undamped. However, real oscillatory processes are damped, because the oscillating body is subject to motion resistance forces (mainly friction forces).
· Forced are called such oscillations, during which the oscillating system is exposed to an external periodically changing force (for example, oscillations of a bridge that occur when people walk along it, walking in step). In many cases, systems undergo oscillations that can be considered harmonic.
· Self-oscillations , like forced oscillations, they are accompanied by an impact on the oscillating system external forces However, the moments of time when these influences are carried out are set by the oscillating system itself. That is, the system itself controls external influences. An example of a self-oscillating system is a clock in which the pendulum receives shocks due to the energy of a raised weight or a twisted spring, and these shocks occur at the moments when the pendulum passes through the middle position.
· Parametric oscillations occur when the parameters of the oscillating system periodically change (a person swinging on a swing periodically raises and lowers his center of gravity, thereby changing the parameters of the system). Under certain conditions, the system becomes unstable - a random deviation from the equilibrium position leads to the emergence and increase of oscillations. This phenomenon is called parametric excitation of oscillations (i.e., oscillations are excited by changing the parameters of the system), and the oscillations themselves are called parametric.
Despite their different physical nature, vibrations are characterized by the same patterns, which are studied by general methods. An important kinematic characteristic is the shape of the vibrations. It is determined by the type of the time function that describes the change in one or another physical quantity during oscillations. The most important fluctuations are those in which the fluctuating quantity changes over time. according to the law of sine or cosine . They're called harmonic .
Harmonic vibrations are called oscillations in which the oscillating physical quantity changes according to the law of sine (or cosine).
This type of oscillation is especially important for the following reasons. Firstly, vibrations in nature and technology often have a character very close to harmonic. Secondly, periodic processes of a different form (with a different time dependence) can be represented as a superposition, or superposition, of harmonic oscillations.
Harmonic Oscillator Equation
Harmonic oscillation is described by a periodic law:
Rice. 18.1. Harmonic oscillation
Z
here
- characterizes change
any physical quantity during oscillations (displacement of the position of the pendulum from the equilibrium position; voltage on the capacitor in the oscillatory circuit, etc.), A
- vibration amplitude
,
-
oscillation phase
,
-
initial phase
,
-
cyclic frequency
; size 
also called
own
vibration frequency. This name emphasizes that this frequency is determined by the parameters of the oscillatory system. A system whose law of motion has the form (18.1) is called one-dimensional harmonic oscillator
. In addition to the listed quantities, the concepts of period
, i.e. time of one oscillation.
(Oscillation period T is the shortest period of time after which the states of the oscillating system are repeated (one complete oscillation is completed) and the oscillation phase receives an increment of 2p).
And frequencies
, which determines the number of oscillations per unit time. The unit of frequency is the frequency of such an oscillation, the period of which is 1 s. This unit is called hertz
(Hz
).
Oscillation frequencyn is the reciprocal of the period of oscillation - the number of complete oscillations performed per unit time.
Amplitude- the maximum value of displacement or change in a variable during oscillatory or wave motion.
Oscillation phase- argument of a periodic function or one describing a harmonic oscillatory process (ω - angular frequency, t- time, - initial phase of oscillations, that is, the phase of oscillations at the initial moment of time t = 0).
The first and second time derivatives of a harmonically oscillating quantity also perform harmonic oscillations of the same frequency:
In this case, the equation of harmonic oscillations written according to the cosine law is taken as a basis. In this case, the first of equations (18.2) describes the law according to which the speed of an oscillating material point (body) changes, the second equation describes the law according to which the acceleration of an oscillating point (body) changes.
Amplitudes 
And 
are equal respectively 
And 
. Hesitation 
ahead 
in phase by ; and hesitation 
ahead 
on 
. Values A And 
can be determined from given initial conditions 
And 
:
,
. (18.3)
Energy of oscillator oscillations
P Rice. 18.2. Spring pendulum
Let's now see what will happen to vibration energy
.
As an example of harmonic oscillations, consider one-dimensional oscillations performed by a body of mass m
Under the influence elastic
strength
(for example, a spring pendulum, see Fig. 18.2). Forces of a different nature than elastic, but in which the condition F = -kx is satisfied, are called quasi-elastic. Under the influence of these forces, bodies also perform harmonic vibrations. Let be:
|
bias: |
|
|
speed: |
|
|
acceleration: |
|
Those. the equation of such oscillations has the form (18.1) with natural frequency 
. The quasi-elastic force is conservative
.
Therefore, the total energy of such harmonic oscillations must remain constant. During the process of oscillations, kinetic energy is converted E To into potential E P and vice versa, and at the moments of the greatest deviation from the equilibrium position, the total energy is equal to the maximum value of the potential energy, and when the system passes through the equilibrium position, the total energy is equal to the maximum value of the kinetic energy. Let's find out how kinetic and potential energy changes over time:
Kinetic energy:
|
|
Potential energy:
(18.5)
Considering that i.e. , the last expression can be written as:
Thus, the total energy of the harmonic oscillation turns out to be constant. From relations (18.4) and (18.5) it also follows that the average values of kinetic and potential energy are equal to each other and half of the total energy, since the average values 
And 
per period are equal to 0.5. Using trigonometric formulas, we can find that kinetic and potential energy change with frequency 
, i.e. with a frequency twice the frequency of harmonic vibration.
Examples of a harmonic oscillator include spring pendulums, physical pendulums, mathematical pendulums, and torsion pendulums.
1. Spring pendulum- this is a load of mass m, which is suspended on an absolutely elastic spring and performs harmonic oscillations under the action of an elastic force F = –kx, where k is the spring stiffness. The equation of motion of the pendulum has the form or (18.8) From formula (18.8) it follows that the spring pendulum performs harmonic oscillations according to the law x = Асos(ω 0 t+φ) with a cyclic frequency
(18.9) and period
(18.10) Formula (18.10) is true for elastic vibrations within the limits within which Hooke’s law is satisfied, that is, if the mass of the spring is small compared to the mass of the body. The potential energy of a spring pendulum, using (18.9) and the potential energy formula of the previous section, is equal to (see 18.5)
2.
Physical pendulum- This solid, which oscillates under the influence of gravity around a fixed horizontal axis that passes through point O, which does not coincide with the center of mass C of the body (Fig. 1).
Fig. 18.3 Physical pendulum
If the pendulum is deflected from the equilibrium position by a certain angle α, then, using the equation of dynamics of the rotational motion of a rigid body, the moment M of the restoring force (18.11) where J is the moment of inertia of the pendulum relative to the axis that passes through the suspension point O, l is the distance between the axis and center of mass of the pendulum, F τ ≈ –mgsinα ≈ –mgα is the restoring force (the minus sign indicates that the directions of F τ and α are always opposite; sinα ≈ α since the oscillations of the pendulum are considered small, i.e. the pendulum is deflected from the equilibrium position by small angles). We write equation (18.11) as
Or Taking (18.12) we obtain the equation
Identical to (18.8), the solution of which will be found and written as:
(18.13) From formula (18.13) it follows that for small oscillations the physical pendulum performs harmonic oscillations with a cyclic frequency ω 0 and a period
(18.14) where the value L=J/(m l) - . Point O" on the continuation of straight line OS, which is located at a distance of reduced length L from the point O of the pendulum suspension, is called swing center physical pendulum (Fig. 18.3). Applying Steiner's theorem for the moment of inertia of the axis, we find
That is, OO" is always greater than OS. The suspension point O of the pendulum and the center of swing O" have interchangeability property: if the suspension point is moved to the center of swing, then the previous suspension point O will be the new center of swing, and the period of oscillation of the physical pendulum will not change.
3. Math pendulum is an idealized system consisting of a material point of mass m, which is suspended on an inextensible weightless thread, and which oscillates under the influence of gravity. A good approximation of a mathematical pendulum is a small heavy ball that is suspended on a long thin thread. Moment of inertia of a mathematical pendulum
(8) where l- length of the pendulum.
Since a mathematical pendulum is a special case of a physical pendulum, if we assume that all its mass is concentrated at one point - the center of mass, then, by substituting (8) into (7), we find an expression for the period of small oscillations of a mathematical pendulum (18.15) Comparing formulas (18.13 ) and (18.15), we see that if the reduced length L of the physical pendulum is equal to the length l mathematical pendulum, then the periods of oscillation of these pendulums are the same. Means, reduced length of a physical pendulum- this is the length of a mathematical pendulum whose period of oscillation coincides with the period of oscillation of a given physical pendulum. For a mathematical pendulum (a material point with mass m, suspended on a weightless inextensible thread of length l in a gravity field with free fall acceleration equal to g) at small angles of deviation (not exceeding 5-10 angular degrees) from the equilibrium position, the natural frequency of oscillations: 
.
4. A body suspended on an elastic thread or other elastic element, oscillating in a horizontal plane, is torsional pendulum.
This is a mechanical oscillatory system that uses elastic deformation forces. In Fig. Figure 18.4 shows the angular analogue of a linear harmonic oscillator performing torsional oscillations. A horizontally located disk hangs on an elastic thread attached to its center of mass. When the disk is rotated through an angle θ, a moment of force occurs M control of elastic torsional deformation:
Where I = IC is the moment of inertia of the disk relative to the axis, passing through the center of mass, ε is the angular acceleration.
By analogy with a load on a spring, you can get.
