Harmonic vibrations occur according to the law. Harmonic vibrations

(lat. amplitude- magnitude) is the greatest deviation of an oscillating body from its equilibrium position.

For a pendulum, this is the maximum distance that the ball moves away from its equilibrium position (figure below). For oscillations with small amplitudes, such a distance can be taken as the length of the arc 01 or 02, and the lengths of these segments.

The amplitude of oscillations is measured in units of length - meters, centimeters, etc. On the oscillation graph, the amplitude is defined as the maximum (modulo) ordinate of the sinusoidal curve (see figure below).

Oscillation period.

Oscillation period- this is the shortest period of time through which a system oscillating returns again to the same state in which it was at the initial moment of time, chosen arbitrarily.

In other words, the oscillation period ( T) is the time during which one complete oscillation occurs. For example, in the figure below, this is the time it takes for the pendulum bob to move from the rightmost point through the equilibrium point ABOUT to the far left point and back through the point ABOUT again to the far right.

Over a full period of oscillation, the body thus travels a path equal to four amplitudes. The period of oscillation is measured in units of time - seconds, minutes, etc. The period of oscillation can be determined from a well-known graph of oscillations (see figure below).

The concept of “oscillation period”, strictly speaking, is valid only when the values ​​of the oscillating quantity are exactly repeated after a certain period of time, i.e. for harmonic oscillations. However, this concept also applies to cases of approximately repeating quantities, for example, for damped oscillations.

Oscillation frequency.

Oscillation frequency- this is the number of oscillations performed per unit of time, for example, in 1 s.

The SI unit of frequency is named hertz(Hz) in honor of the German physicist G. Hertz (1857-1894). If the oscillation frequency ( v) is equal to 1 Hz, this means that every second there is one oscillation. The frequency and period of oscillations are related by the relations:

In the theory of oscillations they also use the concept cyclical, or circular frequency ω . It is related to the normal frequency v and oscillation period T ratios:

.

Cyclic frequency is the number of oscillations performed per 2π seconds

Fedun V.I. Lecture notes on physics Mechanical vibrations and waves

Oscillations and elastic waves.

Lecture 9.

8 . Harmonic oscillations and their main characteristics.

8. 1. Free vibrations and their main characteristics. Representations of harmonic vibrations.

Oscillatory movement is a movement that has some degree of repeatability over time. The movement is called periodic , if the values ​​of quantities change during the movement. Repeated at regular intervals.

A system that oscillates, regardless of its nature, is called oscillator .

Free vibrations. For a body (point) oscillating, there is a position of stable equilibrium. You can remove the body from this state by applying an external force. A body, removed from a state of equilibrium and presented to itself, oscillates around the equilibrium position. Such oscillations are calledown orfree . The frequency with which the system makes such oscillations is called own.

The analytical representation of harmonic oscillations is no less well known:

Where - fluctuating quantity (displacement, speed, acceleration, force, etc.), - time, - amplitude of oscillation (amplitude is equal to the maximum absolute value of the deviation of the oscillating quantity from the equilibrium position), - cyclic or circular frequency.

The physical meaning of the cyclic frequency is that it is numerically equal to the number of oscillations performed per
seconds, i.e.

Where - oscillation frequency, i.e. the number of oscillations performed per unit of time, - period of oscillation - the time during which one complete oscillation occurs.

Magnitude
called the oscillation phase. Oscillation phase is a function of time that determines the value of the oscillating quantity at this moment in time . It shows what part of the amplitude is the offset at a given time:
. Magnitude called the initial phase of oscillation. It determines the value of the quantity at the initial moment of time
.

Finally, in the vector representation, the oscillation is represented as a vector, the length of which is proportional to the amplitude of the oscillations (see Fig. 8.2). The vector itself rotates in the drawing plane with angular velocity  around an axis perpendicular to this plane and passing through the origin of the vector

8. 2. Spring pendulum. Differential equation of free vibrations.

Spring pendulum. An example of a harmonic oscillator is a spring pendulum. Spring pendulum - is a load of mass T, attached to an absolutely elastic spring (see Fig. 8.3) and performing harmonic oscillations under the action of an elastic or quasi-elastic force (forces of a different nature than elastic forces, but also

This force is called restoring force. According to Newton's second law for the restoring force we have

we get differential equation of natural oscillations

or after the transition from complex form to trigonometric

8. 3. 1. Physical pendulum.

Let us consider the rotation of a massive body (see Fig. 8.4) around a fixed axis with small deviations of this body from the equilibrium position. In this case, such a body is called physical pendulum . The equation of motion of this body is the basic equation of the dynamics of rotational motion

Where - moment of inertia of the body, calculated relative to the axis of rotation,
- main vector of moments of forces, - body rotation angle,
- angular acceleration of the body.

Let us remind you that is a pseudovector that is directed along the axis of rotation and obeys the right-hand screw rule. Therefore, in Fig. 8.4 vector directed beyond the plane of the drawing.

Only the moment of gravity causes a body to rotate
, the point of application of which coincides with the center of inertia of the body. Therefore, the main vector of moments of forces

Where
- distance from the axis of rotation to the center of inertia,
-body mass

Projecting (9) onto the axis of rotation, we obtain the differential equation of oscillations of a physical pendulum

8. 3. 2. Mathematical pendulum.

If the dimensions of the body are much less than the distance from the axis of rotation to the center of inertia, then a physical pendulum can be considered mathematical . Here we will check the validity of this statement by passing to the limit for the oscillation frequency, determined by (8.13).

According to Steiner's theorem, the moment of inertia
, - moment of inertia about an axis passing through the center of inertia. If big enough then
. Hence,

8. 4. Speed ​​and acceleration of a body participating in harmonic vibrations. Kinetic, potential and total mechanical energy of the oscillator.

Using the example of a spring pendulum, we will find the speed and acceleration of a body performing an oscillatory motion. By definition, the speed of a body
. Therefore, for harmonic vibrations according to (8.1)

Then, according to the basic law of dynamics, the restoring force

Potential energy of a deformed spring

As you study this section, please keep in mind that fluctuations of different physical nature are described from common mathematical positions. Here it is necessary to clearly understand such concepts as harmonic oscillation, phase, phase difference, amplitude, frequency, oscillation period.

It must be borne in mind that in any real oscillatory system there is resistance of the medium, i.e. the oscillations will be damped. To characterize the damping of oscillations, a damping coefficient and a logarithmic damping decrement are introduced.

If oscillations occur under the influence of an external, periodically changing force, then such oscillations are called forced. They will be undamped. The amplitude of forced oscillations depends on the frequency of the driving force. As the frequency of forced oscillations approaches the frequency of natural oscillations, the amplitude of forced oscillations increases sharply. This phenomenon is called resonance.

When moving on to the study of electromagnetic waves, you need to clearly understand thatelectromagnetic waveis an electromagnetic field propagating in space. The simplest system emitting electromagnetic waves is an electric dipole. If a dipole undergoes harmonic oscillations, then it emits a monochromatic wave.

Formula table: oscillations and waves

Oscillations movements or processes that are characterized by a certain repeatability over time are called. Oscillations are widespread in the surrounding world and can have a very different nature. These can be mechanical (pendulum), electromagnetic (oscillatory circuit) and other types of vibrations.
Free, or own oscillations are called oscillations that occur in a system left to itself, after it has been brought out of equilibrium by an external influence. An example is the oscillation of a ball suspended on a thread.

Special role in oscillatory processes has the simplest form of oscillations - harmonic vibrations. Harmonic oscillations form the basis of a unified approach to the study of oscillations of various natures, since oscillations found in nature and technology are often close to harmonic, and periodic processes of a different form can be represented as a superposition of harmonic oscillations.

Harmonic vibrations are called such oscillations in which the oscillating quantity changes with time according to the law sine or cosine.

Harmonic Equationhas the form:

where A - vibration amplitude (the magnitude of the greatest deviation of the system from the equilibrium position); -circular (cyclic) frequency. The periodically changing argument of the cosine is called oscillation phase . The oscillation phase determines the displacement of the oscillating quantity from the equilibrium position at a given time t. The constant φ represents the phase value at time t = 0 and is called initial phase of oscillation . The value of the initial phase is determined by the choice of the reference point. The x value can take values ​​ranging from -A to +A.

The time interval T through which certain states of the oscillatory system are repeated, called the period of oscillation . Cosine is a periodic function with a period of 2π, therefore, during the period of time T, after which the oscillation phase will receive an increment equal to 2π, the state of the system performing harmonic oscillations will repeat. This period of time T is called the period of harmonic oscillations.

The period of harmonic oscillations is equal to : T = 2π/ .

The number of oscillations per unit time is called vibration frequency ν.
Harmonic frequency is equal to: ν = 1/T. Frequency unit hertz(Hz) - one oscillation per second.

Circular frequency = 2π/T = 2πν gives the number of oscillations in 2π seconds.

Graphically, harmonic oscillations can be depicted as a dependence of x on t (Fig. 1.1.A), and rotating amplitude method (vector diagram method)(Fig.1.1.B) .

The rotating amplitude method allows you to visualize all the parameters included in the harmonic vibration equation. Indeed, if the amplitude vector A located at an angle φ to the x-axis (see Figure 1.1. B), then its projection onto the x-axis will be equal to: x = Acos(φ). The angle φ is the initial phase. If the vector A bring into rotation with an angular velocity equal to the circular frequency of oscillations, then the projection of the end of the vector will move along the x axis and take values ​​ranging from -A to +A, and the coordinate of this projection will change over time according to the law:
.

Thus, the length of the vector is equal to the amplitude of the harmonic oscillation, the direction of the vector at the initial moment forms an angle with the x axis equal to the initial phase of the oscillations φ, and the change in the direction angle with time is equal to the phase of the harmonic oscillations. The time during which the amplitude vector makes one full revolution is equal to the period T of harmonic oscillations. The number of vector revolutions per second is equal to the oscillation frequency ν.

Oscillatory motion- periodic or almost periodic movement of a body, the coordinate, speed and acceleration of which at equal intervals of time take on approximately the same values.

Mechanical vibrations occur when, when a body is removed from an equilibrium position, a force appears that tends to return the body back.

Displacement x is the deviation of the body from the equilibrium position.

Amplitude A is the module of the maximum displacement of the body.

Oscillation period T - time of one oscillation:

Oscillation frequency

The number of oscillations performed by a body per unit of time: During oscillations, the speed and acceleration periodically change. In the equilibrium position, the speed is maximum and the acceleration is zero. At the points of maximum displacement, the acceleration reaches a maximum and the speed becomes zero.

HARMONIC VIBRATION SCHEDULE

Harmonic vibrations that occur according to the law of sine or cosine are called:

where x(t) is the displacement of the system at time t, A is the amplitude, ω is the cyclic frequency of oscillations.

If you plot the deviation of the body from the equilibrium position along the vertical axis, and time along the horizontal axis, you will get a graph of oscillation x = x(t) - the dependence of the body’s displacement on time. For free harmonic oscillations, it is a sine wave or cosine wave. The figure shows graphs of the dependence of displacement x, projections of velocity V x and acceleration a x on time.

As can be seen from the graphs, at maximum displacement x, the speed V of the oscillating body is zero, the acceleration a, and therefore the force acting on the body, is maximum and directed opposite to the displacement. In the equilibrium position, the displacement and acceleration become zero, and the speed is maximum. The acceleration projection always has the opposite sign to the displacement.

ENERGY OF VIBRATIONAL MOTION

The total mechanical energy of an oscillating body is equal to the sum of its kinetic and potential energies and, in the absence of friction, remains constant:

At the moment when the displacement reaches a maximum x = A, the speed, and with it the kinetic energy, goes to zero.

In this case, the total energy is equal to the potential energy:

The total mechanical energy of an oscillating body is proportional to the square of the amplitude of its oscillations.

When the system passes the equilibrium position, the displacement and potential energy are zero: x = 0, E p = 0. Therefore, the total energy is equal to the kinetic energy:

The total mechanical energy of an oscillating body is proportional to the square of its speed in the equilibrium position. Hence:

MATHEMATICAL PENDULUM

1. Math pendulum is a material point suspended on a weightless inextensible thread.

In the equilibrium position, the force of gravity is compensated by the tension of the thread. If the pendulum is deflected and released, then the forces will cease to compensate each other, and a resultant force will arise directed towards the equilibrium position. Newton's second law:

For small oscillations, when the displacement x is much less than l, the material point will move almost along the horizontal x axis. Then from the triangle MAB we get:

Because sin a = x/l, then the projection of the resulting force R onto the x axis is equal to

The minus sign shows that the force R is always directed opposite the displacement x.

2. So, during oscillations of a mathematical pendulum, as well as during oscillations of a spring pendulum, the restoring force is proportional to the displacement and is directed in the opposite direction.

Let's compare the expressions for the restoring force of mathematical and spring pendulums:

It can be seen that mg/l is an analogue of k. Replacing k with mg/l in the formula for the period of a spring pendulum

we obtain the formula for the period of a mathematical pendulum:

The period of small oscillations of a mathematical pendulum does not depend on the amplitude.

A mathematical pendulum is used to measure time and determine the acceleration of gravity at a given location on the earth's surface.

Free oscillations of a mathematical pendulum at small angles of deflection are harmonic. They occur due to the resultant force of gravity and the tension force of the thread, as well as the inertia of the load. The resultant of these forces is the restoring force.

Example. Determine the acceleration due to gravity on a planet where a pendulum 6.25 m long has a period of free oscillation of 3.14 s.

The period of oscillation of a mathematical pendulum depends on the length of the thread and the acceleration of gravity:

By squaring both sides of the equality, we get:

Answer: the acceleration of gravity is 25 m/s 2 .

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