In the equation of harmonic vibration, the quantity under the cosine sign is called. Equation of harmonic vibrations In the equation of harmonic vibration x acos

Oscillations movements or processes that are characterized by a certain repeatability over time are called. Oscillatory processes are widespread in nature and technology, for example, the swinging of a clock pendulum, alternating electric current, etc. When the pendulum oscillates, the coordinate of its center of mass changes; in the case of alternating current, the voltage and current in the circuit fluctuate. The physical nature of vibrations can be different, therefore, there are mechanical, electromagnetic, etc. vibrations. However, different oscillatory processes are described by the same characteristics and the same equations. Hence the expediency common approach to the study of vibrations of different physical nature.

Oscillations are called free, if they occur only under the influence of internal forces acting between the elements of the system, after the system is taken out of equilibrium by external forces and left to itself. Free vibrations always damped oscillations , because in real systems energy losses are inevitable. In the idealized case of a system without energy loss, free oscillations (continuing as long as desired) are called own.

The simplest type of free undamped oscillations are harmonic vibrations - oscillations in which the oscillating quantity changes over time according to the law of sine (cosine). Vibrations found in nature and technology often have a character close to harmonic.

Harmonic oscillations are described by an equation called the harmonic oscillation equation:

Where A- amplitude of oscillations, maximum value of the oscillating quantity X; - circular (cyclic) frequency of natural oscillations; - initial phase of oscillation at the moment of time t= 0; - phase of oscillation at the moment of time t. The oscillation phase determines the value of the oscillating quantity at a given time. Since the cosine varies from +1 to -1, then X can take values ​​from + A before - A.

Time T during which the system completes one complete oscillation is called period of oscillation. During T the oscillation phase is incremented by 2 π , i.e.

Where . (14.2)

The reciprocal of the oscillation period

i.e., the number of complete oscillations performed per unit time is called the oscillation frequency. Comparing (14.2) and (14.3) we get

The unit of frequency is hertz (Hz): 1 Hz is the frequency at which one complete oscillation occurs in 1 s.

Systems in which free vibrations can occur are called oscillators . What properties must a system have in order for free vibrations to occur in it? The mechanical system must have stable equilibrium position, upon exiting which appears restoring force directed towards the equilibrium position. This position corresponds, as is known, to the minimum potential energy of the system. Let us consider several oscillatory systems that satisfy the listed properties.

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).

Harmonic oscillations are oscillations in which a physical quantity changes over time according to a harmonic (sine, cosine) law. The harmonic vibration equation can be written as follows:
X(t) = A∙cos(ω t+φ )
or
X(t) = A∙sin(ω t+φ )

X - deviation from the equilibrium position at time t
A - vibration amplitude, dimension A coincides with dimension X
ω - cyclic frequency, rad/s (radians per second)
φ - initial phase, rad
t - time, s
T - oscillation period, s
f - oscillation frequency, Hz (Hertz)
π is a constant approximately equal to 3.14, 2π=6.28

The oscillation period, frequency in hertz and cyclic frequency are related by relationships.
ω=2πf , T=2π/ω , f=1/T , f=ω/2π
To remember these relationships you need to understand the following.
Each of the parameters ω, f, T uniquely determines the others. To describe the oscillations, it is enough to use one of these parameters.

Period T is the time of one oscillation; it is convenient to use for plotting oscillation graphs.
Cyclic frequency ω - used to write equations of oscillations, allows for mathematical calculations.
Frequency f is the number of oscillations per unit time, used everywhere. In hertz, we measure the frequency to which radios are tuned, as well as the operating range of mobile phones. The frequency of vibration of strings when tuning musical instruments is measured in hertz.

The expression (ωt+φ) is called the oscillation phase, and the value φ is called the initial phase, since it is equal to the oscillation phase at time t=0.

The sine and cosine functions describe the ratios of the sides in a right triangle. Therefore, many do not understand how these functions are related to harmonic vibrations. This relationship is demonstrated by a uniformly rotating vector. The projection of a uniformly rotating vector performs harmonic oscillations.
The picture below shows an example of three harmonic oscillations. Equal in frequency, but different in phase and amplitude.

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 type of periodic oscillations can be accurately represented as a sum of harmonic oscillations, 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, $C=2.5\cdot (10)^(-4)F.$

§ 6. MECHANICAL VIBRATIONSBasic formulas

Harmonic Equation

Where X - displacement of the oscillating point from the equilibrium position; t- time; A,ω, φ - amplitude, angular frequency, initial phase of oscillations, respectively; - phase of oscillations at the moment t.

Angular frequency

where ν and T are the frequency and period of oscillations.

The speed of a point performing harmonic oscillations is

Acceleration during harmonic oscillation

Amplitude A the resulting oscillation obtained by adding two oscillations with the same frequencies, occurring along one straight line, is determined by the formula

Where a 1 And A 2 - amplitudes of vibration components; φ 1 and φ 2 are their initial phases.

The initial phase φ of the resulting oscillation can be found from the formula

The frequency of beats that arise when adding two oscillations occurring along one straight line with different but similar frequencies ν 1 and ν 2,

Equation of the trajectory of a point participating in two mutually perpendicular oscillations with amplitudes A 1 and A 2 and initial phases φ 1 and φ 2,

If the initial phases φ 1 and φ 2 of the oscillation components are the same, then the trajectory equation takes the form

that is, the point moves in a straight line.

In the event that the phase difference is , the equation takes the form

that is, the point moves along an ellipse.

Differential equation of harmonic oscillations of a material point

, or ,where m is the mass of the point; k- quasi-elastic force coefficient ( k=Tω 2).

The total energy of a material point performing harmonic oscillations is

The period of oscillation of a body suspended on a spring (spring pendulum)

Where m- body mass; k- spring stiffness. The formula is valid for elastic vibrations within the limits in which Hooke's law is satisfied (with a small mass of the spring compared to the mass of the body).

Period of oscillation of a mathematical pendulum

Where l- length of the pendulum; g- acceleration of gravity. Period of oscillation of a physical pendulum

Where J- moment of inertia of the oscillating body relative to the axis

hesitation; A- distance of the center of mass of the pendulum from the axis of oscillation;

Reduced length of a physical pendulum.

The given formulas are accurate for the case of infinitesimal amplitudes. For finite amplitudes, these formulas give only approximate results. With amplitudes no greater than, the error in the period value does not exceed 1%.

The period of torsional vibrations of a body suspended on an elastic thread is

Where J- moment of inertia of the body relative to the axis coinciding with the elastic thread; k- the rigidity of an elastic thread, equal to the ratio of the elastic moment arising when the thread is twisted to the angle at which the thread is twisted.

Differential equation of damped oscillations , or ,

Where r- resistance coefficient; δ - damping coefficient: ;ω 0 - natural angular frequency of oscillations *

Damped Oscillation Equation

Where A(t)- amplitude of damped oscillations at the moment t;ω is their angular frequency.

Angular frequency of damped oscillations

О Dependence of the amplitude of damped oscillations on time

I

Where A 0 - amplitude of oscillations at moment t=0.

Logarithmic oscillation decrement

Where A(t) And A(t+T)- amplitudes of two successive oscillations separated in time by a period.

Differential equation of forced oscillations

where is an external periodic force acting on an oscillating material point and causing forced oscillations; F 0 - its amplitude value;

Amplitude of forced oscillations

Resonant frequency and resonant amplitude And

Examples of problem solving

Example 1. The point oscillates according to the law x(t)=, Where A=2 see Determine the initial phase φ if

x(0)=cm and X , (0)<0. Построить векторную диаграмму для мо-­ мента t=0.

Solution. Let's use the equation of motion and express the displacement at the moment t=0 through the initial phase:

From here we find the initial phase:

* In the previously given formulas for harmonic vibrations, the same quantity was designated simply ω (without the index 0).

Let's substitute the given values ​​into this expression x(0) and A:φ= = . The value of the argument is satisfied by two angle values:

In order to decide which of these values ​​of the angle φ also satisfies the condition , we first find:

Substituting the value into this expression t=0 and alternately the values ​​of the initial phases and, we find

T like always A>0 and ω>0, then only the first value of the initial phase satisfies the condition. Thus, the desired initial phase

Using the found value of φ, we construct a vector diagram (Fig. 6.1). Example 2. Material point with mass T=5 g performs harmonic oscillations with frequency ν =0.5 Hz. Oscillation amplitude A=3 cm. Determine: 1) speed υ points at the time when the displacement x== 1.5 cm; 2) the maximum force F max acting on the point; 3) Fig. 6.1 total energy E oscillating point.

and we obtain the speed formula by taking the first time derivative of the displacement:

To express speed through displacement, it is necessary to exclude time from formulas (1) and (2). To do this, we square both equations and divide the first by A 2 , the second one on A 2 ω 2 and add:

, or

Having solved the last equation for υ , we'll find

Having performed calculations using this formula, we get

The plus sign corresponds to the case when the direction of the velocity coincides with the positive direction of the axis X, minus sign - when the direction of velocity coincides with the negative direction of the axis X.

The displacement during harmonic oscillation, in addition to equation (1), can also be determined by the equation

Repeating the same solution with this equation, we get the same answer.

2. We find the force acting on a point using Newton’s second law:

Where A - acceleration of the point, which we obtain by taking the time derivative of the speed:

Substituting the acceleration expression into formula (3), we obtain

Hence the maximum value of force

Substituting the values ​​of π, ν into this equation, T And A, we'll find

3. The total energy of an oscillating point is the sum of the kinetic and potential energies calculated for any moment in time.

The easiest way to calculate the total energy is at the moment when the kinetic energy reaches its maximum value. At this moment the potential energy is zero. Therefore the total energy E the oscillating point is equal to the maximum kinetic energy

We determine the maximum speed from formula (2), putting: . Substituting the expression for speed into formula (4), we find

Substituting the values ​​of quantities into this formula and making calculations, we get

or µJ.

Example 3. At the ends of a thin rod length l= 1 m and mass m 3 =400 g reinforced small balls with masses m 1 =200 g And m 2 =300g. The rod oscillates about a horizontal axis, perpendicular

dicular to the rod and passing through its middle (point O in Fig. 6.2). Define period T oscillations made by the rod.

Solution. The period of oscillation of a physical pendulum, such as a rod with balls, is determined by the relation

Where J- T - its mass; l WITH - the distance from the center of mass of the pendulum to the axis.

The moment of inertia of this pendulum is equal to the sum of the moments of inertia of the balls J 1 and J 2 and rod J 3:

Taking the balls as material points, we express their moments of inertia:

Since the axis passes through the middle of the rod, its moment of inertia relative to this axis J 3 = =. Substituting the resulting expressions J 1 , J 2 And J 3 into formula (2), we find the total moment of inertia of the physical pendulum:

Having carried out calculations using this formula, we find

Rice. 6.2 The mass of the pendulum consists of the masses of the balls and the mass of the rod:

Distance l WITH We will find the center of mass of the pendulum from the axis of oscillation based on the following considerations. If the axis X direct along the rod and align the origin of coordinates with the point ABOUT, then the required distance l equal to the coordinate of the center of mass of the pendulum, i.e.

Substituting the values ​​of the quantities m 1 , m 2 , m, l and after performing calculations, we find

Having made calculations using formula (1), we obtain the oscillation period of a physical pendulum:

Example 4. A physical pendulum is a rod of length l= 1 m and mass 3 T 1 With attached to one of its ends with a hoop of diameter and mass T 1 . Horizontal axis Oz

the pendulum passes through the middle of the rod perpendicular to it (Fig. 6.3). Define period T oscillations of such a pendulum.

Solution. The period of oscillation of a physical pendulum is determined by the formula

(1)

Where J- moment of inertia of the pendulum relative to the axis of oscillation; T - its mass; l C - the distance from the center of mass of the pendulum to the axis of oscillation.

The moment of inertia of the pendulum is equal to the sum of the moments of inertia of the rod J 1 and hoop J 2:

(2).

The moment of inertia of the rod relative to the axis perpendicular to the rod and passing through its center of mass is determined by the formula . In this case t= 3T 1 and

We find the moment of inertia of the hoop using Steiner’s theorem ,Where J- moment of inertia about an arbitrary axis; J 0 - moment of inertia about an axis passing through the center of mass parallel to a given axis; A - the distance between the indicated axes. Applying this formula to the hoop, we get

Substituting expressions J 1 and J 2 into formula (2), we find the moment of inertia of the pendulum relative to the axis of rotation:

Distance l WITH from the axis of the pendulum to its center of mass is equal to

Substituting the expressions into formula (1) J, l s and the mass of the pendulum, we find the period of its oscillations:

After calculating using this formula we get T=2.17 s.

Example 5. Two oscillations of the same direction are added, expressed by the equations; X 2 = =, where A 1 = 1 cm, A 2 =2 cm, s, s, ω = =. 1. Determine the initial phases φ 1 and φ 2 of the components of the oscillator

Baniya. 2. Find the amplitude A and the initial phase φ of the resulting oscillation. Write the equation for the resulting vibration.

Solution. 1. The equation of harmonic vibration has the form

Let us transform the equations specified in the problem statement to the same form:

From a comparison of expressions (2) with equality (1), we find the initial phases of the first and second oscillations:

Glad and glad.

2. To determine the amplitude A of the resulting oscillation, it is convenient to use the vector diagram presented in rice. 6.4. According to the cosine theorem, we get

where is the phase difference of the oscillation components. Since , then by substituting the found values ​​of φ 2 and φ 1 we get rad.

Let's substitute the values A 1 , A 2 and into formula (3) and perform the calculations:

A= 2.65 cm.

Let us determine the tangent of the initial phase φ of the resulting oscillation directly from Fig. 6.4: ,where does the initial phase come from?