Oscillation of a load on a spring. Free oscillations of a spring pendulum

I'M IN. ,
Far Eastern State Interregional Industrial and Economic College, Khabarovsk

Body vibrations on a spring

Educational goals: formation of an idea of the process of scientific knowledge, organization and systematization of knowledge on the topic; developing an idea of the dependence of the oscillation period on body weight and spring stiffness; development of experimental skills, research skills.

Equipment: tape recorder, computers, program or (section “Mechanical vibrations and waves”, “Body vibrations on a spring”), § 31 of the textbook.

During the classes

1. Start of class

Teacher (begins the lesson with a poem by B. Pasternak: “In everything I want to get to the very essence<...>//Make the discovery”). What does the words “I made a discovery” mean to you guys? ( Listens to the answers.) Did I understand you correctly: if a person, through his hard work and perseverance, achieves the truth in something, then this means that he has made a discovery? Today we will also make small, but independent discoveries. So, the topic of our lesson is “Body vibrations on a spring.”

2. Repetition and generalization

Teacher. First, let's admire together our deep knowledge on the topic of Mechanical Vibrations. Write down the missing left parts of the formulas in cards ( one student performs a task at the board):

(The class checks its notes, everyone gives themselves points on the self-control sheet according to the number of formulas they wrote correctly and the number of formulas found with errors.)

Now let’s pull out something valuable from the caches of memory. Here is a table with physical quantities, their units, and numbers. I will ask a question, and you will cross out the box with the correct answer:

Time interval during which one complete oscillation occurs Maximum deviation of the oscillating quantity from the equilibrium position Number of oscillations per unit time Unit of oscillation period Unit of oscillation frequency Unit of oscillation amplitude During what time the pendulum completed n= 20 oscillations if the oscillation period is 0.5 s? What is the frequency of these oscillations? The body oscillates along the axis X. Its coordinate changes with time according to the law x= 0.2cos0.63 t(SI). What is the amplitude of the body's vibrations? What is the cyclic frequency of these oscillations? A very soft large spring contracts in 2 s from its maximum stretch to its original state. What is the period of oscillation of the spring? If the length of the spring changes by 0.5 m, what is the distance traveled by the loose end of the spring during the period of oscillation?

(Correct answers “draw” the number “5” on the card. The guys put a mark on the self-control sheet - 1 point for the correct answer.)

The basis of any branch of physics is observation or experiment. Today I invite you to conduct research on mechanical vibrations. Break into four groups as desired. Each group takes a card with a task and completes it, and then tells what they did and what they received.

Task No. 1. Make a seconds pendulum (oscillation period 1 s). Devices and materials: thread, weight, ruler, stopwatch.

Task No. 2. Determine the period of oscillation of a meter-long string pendulum. What will it be equal to if the length of the thread is reduced by four times? Devices and materials: meter pendulum, stopwatch.

Task No. 3. Determine the period, frequency and cyclic frequency of the pendulum's oscillations. Write down the equation of oscillation of this pendulum. Devices and materials: ball, ruler, stopwatch, thread.

Task No. 4. Determine in practice the acceleration of gravity for a given area using a string pendulum. Devices and materials: thread, ball, ruler, stopwatch.

(The teacher evaluates the work of the groups. The guys put points on a self-control sheet: 1 point for conducting an experiment, 1 point for defending.)

3. Learning new material

Teacher. Now let’s move on to the topic of our lesson, “Body oscillations on a spring.” Let's try to establish the dependence of the period of free oscillations on the mass of the load, the stiffness of the spring and the amplitude of oscillations. ( The guys are divided into pairs at will, receive cards, during a computer experiment they establish these dependencies and write down the results and conclusions on the cards. .)

Establish the dependence of the period of free oscillations on the mass and stiffness of the spring

Fill the table

Draw a conclusion: if you increase the spring stiffness, then the period: decreases.

A, cm	5	7	10
T, With	1,4	1,4	1,4

Draw a conclusion: if you increase the amplitude of oscillations, then the period: does not change.

Write down the formula for the period of free oscillations

Use § 38 of the textbook V.A. Kasyanova"Physics-10":

Draw a conclusion: the period of free oscillation of a spring pendulum does not depend on amplitude of oscillations, and is completely determined by rigidity, mass (the own characteristics of the oscillatory system).

Check experimentally the dependence of the period of free oscillations on mass and stiffness.

I would like to guide you in your work with the words of A. Tolstoy: “Knowledge is only knowledge when it is acquired through the efforts of one’s thoughts, and not memory.” Good luck with your research!

(The guys establish dependencies, put 1 point for each formula on the self-control sheet.)

4. Consolidation, training, skill development

Teacher. Now let’s solve the problems on cards and check the answer using a computer experiment. The solution to the first problem is worth a maximum of 1 point, the second – 2 points.

Task 1. Determine the period of oscillation of a spring pendulum if the mass of the load is 0.5 kg and the spring stiffness is 10 N/m.

Task 2. Write the equation of motion of a spring pendulum x(t), If m= 1 kg, k= 10 N/m, A= 10 cm. Determine the coordinate at the moment of time t= 4 s.

Check the answer according to the graph, to do this, select the parameters, click Start and follow the readings t.

Creative task. Come up with, formulate and solve a problem, conduct a computer experiment and check your answer. Enter the teacher's assessment (up to 2 points) on the self-control sheet.

5. Reflection. Summarizing

Teacher. Let's summarize. What was the main thing? What was interesting? What new did you learn today? What did you learn? ( Listens to opinions. The guys count the points and give themselves marks: 24–25 points – “3”, 26–27 points – “4”, 28–29 points – “5”.)

DZ.§ 38, tasks 1, 2. Make up your own tasks for future students. Be sure to sign your work, the authorship will be preserved. And I want to end today’s lesson with the words of M. Faraday: “The art of the experimenter is to be able to ask nature questions and understand its answers.” And I think you succeeded today. The lesson is over. Thank you for the lesson. I wish you success. See you in the next lesson.

Literature

Physics in pictures 6.2. NC PHYSIKON, 1993. 1 electron. wholesale disk (DVD-ROM); [Electronic resource] URL: http://torrents.ru/forum/.
Open Physics 2.6: Part 1: LLC FISIKON, 1996–2005 [Electronic resource] URL: http://physics.ru
Kasyanov V.A. Physics: textbook. for general education institutions. 10 grades M.: Bustard, 2003. pp. 123–133.

Yana Vladimirovna Bocharnikova in 1990, she graduated from Far Eastern State University with a degree in physics, physics teacher, worked at the Khabarovsk Institute of Railway Transport Engineers, then taught computer science at a preschool educational institution for children 3–7 years old, taught physics at school and for 9 years now - at college. Winner of the city competition “Teacher of the Year-99” and the competition “Teacher of the Year-2005” in college, laureate of the regional competition “Teacher of the Year-2005”. In his work, he is guided by the words of S. Soloveichik: “To raise people with a deep sense of self-worth, full of self-respect and respect for others, people who are able to choose, to act independently - doesn’t this mean contributing to the strengthening and prosperity of the country?”

Student entries are highlighted here in gray font. – Ed.

Free vibrations are carried out under the influence of internal forces of the system after the system has been removed from its equilibrium position.

In order to free vibrations occur according to the harmonic law, it is necessary that the force tending to return the body to the equilibrium position is proportional to the displacement of the body from the equilibrium position and is directed in the direction opposite to the displacement (see §2.1):

Forces of any other physical nature that satisfy this condition are called quasi-elastic .

Thus, a load of some mass m, attached to the stiffening spring k, the second end of which is fixedly fixed (Fig. 2.2.1), constitute a system capable of performing free harmonic oscillations in the absence of friction. A load on a spring is called linear harmonic oscillator.

The circular frequency ω 0 of free oscillations of a load on a spring is found from Newton’s second law:

When the spring-load system is located horizontally, the force of gravity applied to the load is compensated by the support reaction force. If the load is suspended on a spring, then the force of gravity is directed along the line of movement of the load. In the equilibrium position, the spring is stretched by an amount x 0 equal

Therefore, Newton's second law for a load on a spring can be written as

Equation (*) is called equation of free vibrations . It should be noted that the physical properties of the oscillatory system determine only the natural frequency of oscillations ω 0 or the period T . Parameters of the oscillation process such as amplitude x m and the initial phase φ 0 are determined by the way in which the system was brought out of equilibrium at the initial moment of time.

If, for example, the load was displaced from the equilibrium position by a distance Δ l and then at a point in time t= 0 released without initial speed, then x m = Δ l, φ 0 = 0.

If the load, which was in the equilibrium position, was given an initial speed ± υ 0 with the help of a sharp push, then,

Thus, the amplitude x m free oscillations and its initial phase φ 0 are determined initial conditions .

There are many types of mechanical oscillatory systems that use elastic deformation forces. In Fig. Figure 2.2.2 shows the angular analogue of a linear harmonic oscillator. 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 = I C 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:

Free vibrations. Math pendulum

Mathematical pendulum called a small body suspended on a thin inextensible thread, the mass of which is negligible compared to the mass of the body. In the equilibrium position, when the pendulum hangs plumb, the force of gravity is balanced by the tension force of the thread. When the pendulum deviates from the equilibrium position by a certain angle φ, a tangential component of gravity appears F τ = - mg sin φ (Fig. 2.3.1). The minus sign in this formula means that the tangential component is directed in the direction opposite to the deflection of the pendulum.

If we denote by x linear displacement of the pendulum from the equilibrium position along an arc of a circle of radius l, then its angular displacement will be equal to φ = x / l. Newton's second law, written for the projections of acceleration and force vectors onto the direction of the tangent, gives:

This relationship shows that a mathematical pendulum is a complex nonlinear system, since the force tending to return the pendulum to the equilibrium position is not proportional to the displacement x, A

Only in case small fluctuations, when approximately can be replaced by a mathematical pendulum is a harmonic oscillator, that is, a system capable of performing harmonic oscillations. In practice, this approximation is valid for angles of the order of 15-20°; in this case, the value differs from by no more than 2%. The oscillations of a pendulum at large amplitudes are not harmonic.

For small oscillations of a mathematical pendulum, Newton's second law is written as

This formula expresses natural frequency of small oscillations of a mathematical pendulum .

Hence,

Any body mounted on a horizontal axis of rotation is capable of free oscillations in a gravitational field and, therefore, is also a pendulum. Such a pendulum is usually called physical (Fig. 2.3.2). It differs from the mathematical one only in the distribution of masses. In a stable equilibrium position, the center of mass C the physical pendulum is located below the axis of rotation O on the vertical passing through the axis. When the pendulum is deflected by an angle φ, a moment of gravity arises, tending to return the pendulum to the equilibrium position:

and Newton’s second law for a physical pendulum takes the form (see §1.23)

Here ω 0 - natural frequency of small oscillations of a physical pendulum .

Hence,

Therefore, the equation expressing Newton’s second law for a physical pendulum can be written in the form

Finally, for the circular frequency ω 0 of free oscillations of a physical pendulum, the following expression is obtained:

Energy transformations during free mechanical vibrations

During free mechanical vibrations, kinetic and potential energies change periodically. At the maximum deviation of a body from its equilibrium position, its speed, and therefore its kinetic energy, vanish. In this position, the potential energy of the oscillating body reaches its maximum value. For a load on a spring, potential energy is the energy of elastic deformation of the spring. For a mathematical pendulum, this is the energy in the Earth's gravitational field.

When a body in its motion passes through the equilibrium position, its speed is maximum. The body overshoots the equilibrium position according to the law of inertia. At this moment it has maximum kinetic and minimum potential energy. An increase in kinetic energy occurs due to a decrease in potential energy. With further movement, potential energy begins to increase due to a decrease in kinetic energy, etc.

Thus, during harmonic oscillations, a periodic transformation of kinetic energy into potential energy and vice versa occurs.

If there is no friction in the oscillatory system, then the total mechanical energy during free oscillations remains unchanged.

For spring load(see §2.2):

In real conditions, any oscillatory system is under the influence of friction forces (resistance). In this case, part of the mechanical energy is converted into internal energy of thermal motion of atoms and molecules, and vibrations become fading (Fig. 2.4.2).

The rate at which vibrations decay depends on the magnitude of friction forces. Time interval τ during which the amplitude of oscillations decreases in e≈ 2.7 times, called decay time .

The frequency of free oscillations depends on the rate at which the oscillations decay. As friction forces increase, the natural frequency decreases. However, the change in the natural frequency becomes noticeable only with sufficiently large friction forces, when the natural vibrations quickly decay.

An important characteristic of an oscillatory system performing free damped oscillations is quality factor Q. This parameter is defined as a number N total oscillations performed by the system during the damping time τ, multiplied by π:

Thus, the quality factor characterizes the relative loss of energy in the oscillatory system due to the presence of friction over a time interval equal to one oscillation period.

Forced vibrations. Resonance. Self-oscillations

Oscillations occurring under the influence of an external periodic force are called forced.

An external force does positive work and provides an energy flow to the oscillatory system. It does not allow vibrations to die out, despite the action of friction forces.

A periodic external force can change over time according to various laws. Of particular interest is the case when an external force, varying according to a harmonic law with a frequency ω, acts on an oscillatory system capable of performing its own oscillations at a certain frequency ω 0.

If free oscillations occur at a frequency ω 0, which is determined by the parameters of the system, then steady forced oscillations always occur at frequency ω external force.

After the external force begins to act on the oscillatory system, some time Δ t to establish forced oscillations. The establishment time is, in order of magnitude, equal to the damping time τ of free oscillations in the oscillatory system.

At the initial moment, both processes are excited in the oscillatory system - forced oscillations at frequency ω and free oscillations at natural frequency ω 0. But free vibrations are damped due to the inevitable presence of friction forces. Therefore, after some time, only stationary oscillations at the frequency ω of the external driving force remain in the oscillatory system.

Let us consider, as an example, forced oscillations of a body on a spring (Fig. 2.5.1). An external force is applied to the free end of the spring. It forces the free (left in Fig. 2.5.1) end of the spring to move according to the law

If the left end of the spring is displaced by a distance y, and the right one - to the distance x from their original position, when the spring was undeformed, then the elongation of the spring Δ l equals:

In this equation, the force acting on a body is represented as two terms. The first term on the right side is the elastic force tending to return the body to the equilibrium position ( x= 0). The second term is the external periodic effect on the body. This term is called coercive force.

The equation expressing Newton's second law for a body on a spring in the presence of an external periodic influence can be given a strict mathematical form if we take into account the relationship between the acceleration of the body and its coordinate: Then will be written in the form

Equation (**) does not take into account the action of friction forces. Unlike equations of free vibrations(*) (see §2.2) forced oscillation equation(**) contains two frequencies - the frequency ω 0 of free oscillations and the frequency ω of the driving force.

Steady-state forced oscillations of a load on a spring occur at the frequency of external influence according to the law

x(t) = x mcos(ω t + θ).

Amplitude of forced oscillations x m and the initial phase θ depend on the ratio of frequencies ω 0 and ω and on the amplitude y m external force.

At very low frequencies, when ω<< ω 0 , движение тела массой m, attached to the right end of the spring, repeats the movement of the left end of the spring. Wherein x(t) = y(t), and the spring remains practically undeformed. An external force applied to the left end of the spring does not do any work, since the modulus of this force at ω<< ω 0 стремится к нулю.

If the frequency ω of the external force approaches the natural frequency ω 0, a sharp increase in the amplitude of forced oscillations occurs. This phenomenon is called resonance . Amplitude dependence x m forced oscillations from the frequency ω of the driving force is called resonant characteristic or resonance curve(Fig. 2.5.2).

At resonance, the amplitude x m oscillations of the load can be many times greater than the amplitude y m vibrations of the free (left) end of the spring caused by external influence. In the absence of friction, the amplitude of forced oscillations during resonance should increase without limit. In real conditions, the amplitude of steady-state forced oscillations is determined by the condition: the work of the external force during the oscillation period must be equal to the loss of mechanical energy during the same time due to friction. The less friction (i.e. the higher the quality factor Q oscillatory system), the greater the amplitude of forced oscillations at resonance.

In oscillatory systems with not very high quality factor (< 10) резонансная частота несколько смещается в сторону низких частот. Это хорошо заметно на рис. 2.5.2.

The phenomenon of resonance can cause the destruction of bridges, buildings and other structures if the natural frequencies of their oscillations coincide with the frequency of a periodically acting force, which arises, for example, due to the rotation of an unbalanced motor.

Forced vibrations are undamped fluctuations. The inevitable energy losses due to friction are compensated by the supply of energy from an external source of periodically acting force. There are systems in which undamped oscillations arise not due to periodic external influences, but as a result of the ability of such systems to regulate the supply of energy from a constant source. Such systems are called self-oscillating, and the process of undamped oscillations in such systems is self-oscillations . In a self-oscillating system, three characteristic elements can be distinguished - an oscillatory system, an energy source, and a feedback device between the oscillatory system and the source. Any mechanical system capable of performing its own damped oscillations (for example, the pendulum of a wall clock) can be used as an oscillatory system.

The energy source can be the deformation energy of a spring or the potential energy of a load in a gravitational field. A feedback device is a mechanism by which a self-oscillating system regulates the flow of energy from a source. In Fig. 2.5.3 shows a diagram of the interaction of various elements of a self-oscillating system.

An example of a mechanical self-oscillating system is a clock mechanism with anchor progress (Fig. 2.5.4). The running wheel with oblique teeth is rigidly attached to a toothed drum, through which a chain with a weight is thrown. At the upper end of the pendulum is fixed anchor(anchor) with two plates of solid material, bent in a circular arc with the center on the axis of the pendulum. In hand watches, the weight is replaced by a spring, and the pendulum is replaced by a balancer - a handwheel connected to a spiral spring. The balancer performs torsional vibrations around its axis. The oscillatory system in a clock is a pendulum or balancer.

The source of energy is a raised weight or a wound spring. The device used to provide feedback is an anchor, which allows the running wheel to turn one tooth in one half-cycle. Feedback is provided by the interaction of the anchor with the running wheel. With each oscillation of the pendulum, a tooth of the running wheel pushes the anchor fork in the direction of movement of the pendulum, transferring to it a certain portion of energy, which compensates for energy losses due to friction. Thus, the potential energy of the weight (or twisted spring) is gradually, in separate portions, transferred to the pendulum.

Mechanical self-oscillating systems are widespread in life around us and in technology. Self-oscillations occur in steam engines, internal combustion engines, electric bells, strings of bowed musical instruments, air columns in the pipes of wind instruments, vocal cords when speaking or singing, etc.

Figure 2.5.4. Clock mechanism with a pendulum.

During mechanical vibrations of a body on a spring, the coordinate of the body will periodically change. In this case, the projection of the body’s velocity onto the Ox axis will change. In electromagnetic oscillations, over time, according to a periodic law, the charge q of the capacitor and the current strength in the circuit of the oscillatory circuit will change.

The quantities will have the same pattern of change. This happens because there is an analogy between the conditions in which oscillations occur. When we remove the load on the spring from the equilibrium position, an elastic force F ex. arises in the spring, which tends to return the load back to the equilibrium position. The proportionality coefficient of this force will be the spring stiffness k.

When the capacitor discharges, a current appears in the oscillatory circuit circuit. Discharge is due to the fact that there is a voltage u across the capacitor plates. This voltage will be proportional to the charge q of any of the plates. The proportionality coefficient will be the value 1/C, Where C is the capacitance of the capacitor.

When a load moves on a spring, when we release it, the speed of the body increases gradually, due to inertia. And after the cessation of force, the speed of the body does not immediately become zero, it also gradually decreases.

Mechanical movement. Trajectory, path, movement. Mechanical movement is a change in the position of a body in space relative to other bodies. The trajectory of movement is the line along which the body moves. According to the shape of the trajectory, the movement of bodies can be rectilinear or curvilinear. Rectilinear uniform motion is a movement in which a body travels identical paths for any (!) equal (!) periods of time.

The relativity of motion means that the characteristics of motion (trajectory, path, speed, etc.) depend on the choice of the body of reference. A body of reference is a body relative to which motion is considered. A material point is a body whose dimensions are neglected under given conditions. (body mass is concentrated at this point)

Free vibrations are carried out under the influence of internal forces of the system after the system has been removed from its equilibrium position.

In order for free vibrations to occur according to the harmonic law, it is necessary that the force tending to return the body to the equilibrium position be proportional to the displacement of the body from the equilibrium position and directed in the direction opposite to the displacement:

Forces of any other physical nature that satisfy this condition are called quasi-elastic .

The circular frequency ω 0 of free oscillations of a load on a spring is found from Newton’s second law:

hence

The frequency ω 0 is called natural frequency oscillatory system.

Period T harmonic vibrations of the load on the spring is equal to

and oscillations occur around this new equilibrium position. The above expressions for the natural frequency ω 0 and the oscillation period T are also valid in this case.

A rigorous description of the behavior of the oscillatory system can be given if we take into account the mathematical relationship between the acceleration of the body a and coordinate x: acceleration is the second derivative of the body coordinate x by time t :

Therefore, Newton's second law for a load on a spring can be written as

(*)

All physical systems (not only mechanical) described by equation (*) are capable of performing free harmonic oscillations, since the solution to this equation is harmonic functions of the form

x = x mcos(ω t + φ 0).

Equation (*) is called equation of free vibrations . It should be noted that the physical properties of the oscillatory system determine only the natural frequency of oscillations ω0 or period T . Such parameters of the oscillatory process as amplitude x m and the initial phase φ 0 are determined by the way in which the system was brought out of equilibrium at the initial moment of time.

If, for example, the load was displaced from the equilibrium position by a distance Δ l and then at a point in time t= 0 released without initial speed, then x m = Δ l, φ 0 = 0.

If the load, which was in the equilibrium position, was given an initial speed with the help of a sharp push, then

Thus, the amplitude x m free oscillations and its initial phase φ 0 are determined initial conditions .

There are many types of mechanical oscillatory systems that use elastic deformation forces. In Fig. Figure 2.2.2 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:

This relationship expresses Hooke's law for torsional deformation. The value of χ is similar to the spring stiffness k. Newton's second law for the rotational motion of a disk is written as

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:

The torsion pendulum is widely used in mechanical watches. It is called a balancer. In the balancer, the moment of elastic forces is created using a spiral spring.

In technology and the world around us we often have to deal with periodic(or almost periodic) processes that repeat at regular intervals. Such processes are called oscillatory.

Oscillations are one of the most common processes in nature and technology. The wings of insects and birds in flight, high-rise buildings and high-voltage wires under the influence of the wind, the pendulum of a wound clock and a car on springs while driving, the river level throughout the year and the temperature of the human body during illness, sound is fluctuations in air density and pressure, radio waves - periodic changes in the strengths of electric and magnetic fields, visible light is also electromagnetic vibrations, only with slightly different wavelengths and frequencies, earthquakes are soil vibrations, the pulse is periodic contractions of the human heart muscle, etc.

Oscillations can be mechanical, electromagnetic, chemical, thermodynamic and various others. Despite such diversity, they all have much in common.

Oscillatory phenomena of various physical natures are subject to general laws. For example, current oscillations in an electrical circuit and oscillations of a mathematical pendulum can be described by the same equations. The commonality of oscillatory patterns allows us to consider oscillatory processes of various natures from a single point of view. A sign of oscillatory motion is its periodicity.

Mechanical vibrations –Thismovements that are repeated exactly or approximately at regular intervals.

Examples of simple oscillatory systems are a load on a spring (spring pendulum) or a ball on a string (mathematical pendulum).

During mechanical vibrations, kinetic and potential energies change periodically.

At maximum deviation body from its equilibrium position, its speed, and therefore kinetic energy goes to zero. In this position potential energy oscillating body reaches maximum value. For a load on a spring, potential energy is the energy of elastic deformation of the spring. For a mathematical pendulum, this is the energy in the Earth’s gravitational field.

When a body, in its movement, passes through equilibrium position, its speed is maximum. The body overshoots the equilibrium position according to the law of inertia. At this moment it has maximum kinetic and minimum potential energy. An increase in kinetic energy occurs due to a decrease in potential energy.

With further movement, potential energy begins to increase due to a decrease in kinetic energy, etc.

Thus, during harmonic oscillations, a periodic transformation of kinetic energy into potential energy and vice versa occurs.

If there is no friction in the oscillatory system, then the total mechanical energy during mechanical vibrations remains unchanged.

For spring load:

At the position of maximum deflection, the total energy of the pendulum is equal to the potential energy of the deformed spring:

When passing through the equilibrium position, the total energy is equal to the kinetic energy of the load:

For small oscillations of a mathematical pendulum:

At the position of maximum deviation, the total energy of the pendulum is equal to the potential energy of the body raised to a height h:

When passing through the equilibrium position, the total energy is equal to the kinetic energy of the body:

Here h m– the maximum height of the pendulum in the Earth’s gravitational field, x m and υ m = ω 0 x m– maximum values of the pendulum’s deviation from the equilibrium position and its speed.

Harmonic oscillations and their characteristics. Equation of harmonic vibration.

The simplest type of oscillatory process are simple harmonic vibrations, which are described by the equation

x = x m cos(ω t + φ 0).

Here x– displacement of the body from the equilibrium position,
x m– amplitude of oscillations, that is, the maximum displacement from the equilibrium position,
ω – cyclic or circular frequency hesitation,
t- time.

Characteristics of oscillatory motion.

Offset x – deviation of an oscillating point from its equilibrium position. The unit of measurement is 1 meter.

Oscillation amplitude A – the maximum deviation of an oscillating point from its equilibrium position. The unit of measurement is 1 meter.

Oscillation periodT– the minimum time interval during which one complete oscillation occurs is called. The unit of measurement is 1 second.

T=t/N

where t is the time of oscillations, N is the number of oscillations completed during this time.

From the graph of harmonic oscillations, you can determine the period and amplitude of the oscillations:

Oscillation frequency ν – a physical quantity equal to the number of oscillations per unit of time.

ν=N/t

Frequency is the reciprocal of the oscillation period:

Frequency oscillations ν shows how many oscillations occur in 1 s. The unit of frequency is hertz(Hz).

Cyclic frequency ω– number of oscillations in 2π seconds.

The oscillation frequency ν is related to cyclic frequency ω and oscillation period T ratios:

Phase harmonic process - a quantity under the sine or cosine sign in the equation of harmonic oscillations φ = ω t + φ 0 . At t= 0 φ = φ 0 , therefore φ 0 called initial phase.

Harmonic graph represents a sine or cosine wave.

In all three cases for blue curves φ 0 = 0:

only greater amplitude(x" m > x m);

the red curve is different from the blue one only meaning period(T" = T / 2);

the red curve is different from the blue one only meaning initial phase(glad).

When a body oscillates along a straight line (axis OX) the velocity vector is always directed along this straight line. The speed of movement of the body is determined by the expression

In mathematics, the procedure for finding the limit of the ratio Δх/Δt at Δ t→ 0 is called calculating the derivative of the function x(t) by time t and is denoted as x"(t).The speed is equal to the derivative of the function x( t) by time t.

For the harmonic law of motion x = x m cos(ω t+ φ 0) calculating the derivative leads to the following result:

υ X =x"(t)= ω x m sin (ω t + φ 0)

Acceleration is determined in a similar way a x bodies during harmonic vibrations. Acceleration a is equal to the derivative of the function υ( t) by time t, or the second derivative of the function x(t). Calculations give:

and x =υ x "(t) =x""(t)= -ω 2 x m cos(ω t+ φ 0)=-ω 2 x

The minus sign in this expression means that the acceleration a(t) always has the opposite sign of the displacement x(t), and, therefore, according to Newton’s second law, the force causing the body to perform harmonic oscillations is always directed towards the equilibrium position ( x = 0).

The figure shows graphs of the coordinates, speed and acceleration of a body performing harmonic oscillations.

Graphs of coordinates x(t), speed υ(t) and acceleration a(t) of a body performing harmonic oscillations.

Spring pendulum.

Spring pendulumis a load of some mass m attached to a spring of stiffness k, the second end of which is fixedly fixed.

Natural frequencyω 0 free oscillations of the load on the spring is found by the formula:

Period T harmonic vibrations of the load on the spring is equal to

This means that the period of oscillation of a spring pendulum depends on the mass of the load and the stiffness of the spring.

Physical properties of an oscillatory system determine only the natural frequency of oscillations ω 0 and the period T . Parameters of the oscillation process such as amplitude x m and the initial phase φ 0 are determined by the way in which the system was brought out of equilibrium at the initial moment of time.

Mathematical pendulum.

Mathematical pendulumcalled a small body suspended on a thin inextensible thread, the mass of which is negligible compared to the mass of the body.

In the equilibrium position, when the pendulum hangs plumb, the force of gravity is balanced by the tension force of the thread N. When the pendulum deviates from the equilibrium position by a certain angle φ, a tangential component of the force of gravity appears F τ = – mg sin φ. The minus sign in this formula means that the tangential component is directed in the direction opposite to the deflection of the pendulum.

Mathematical pendulum.φ – angular deviation of the pendulum from the equilibrium position,

x= lφ – displacement of the pendulum along the arc

The natural frequency of small oscillations of a mathematical pendulum is expressed by the formula:

Period of oscillation of a mathematical pendulum:

This means that the period of oscillation of a mathematical pendulum depends on the length of the thread and on the acceleration of free fall of the area where the pendulum is installed.

Free and forced vibrations.

Mechanical vibrations, like oscillatory processes of any other physical nature, can be free And forced.

Free vibrations –These are oscillations that occur in a system under the influence of internal forces, after the system has been removed from a stable equilibrium position.

Oscillations of a weight on a spring or oscillations of a pendulum are free oscillations.

Fading called oscillations whose amplitude decreases with time.

To prevent the oscillations from fading, it is necessary to provide additional energy to the system, i.e. influence the oscillatory system with a periodic force (for example, to rock a swing).

Oscillations occurring under the influence of an external periodically changing force are calledforced.

An external force does positive work and provides an energy flow to the oscillatory system. It does not allow vibrations to die out, despite the action of friction forces.

If free oscillations occur at a frequency ω 0, which is determined by the parameters of the system, then steady forced oscillations always occur at frequency ω external force .

The phenomenon of a sharp increase in the amplitude of forced oscillations when the frequency of natural oscillations coincides with the frequency of the external driving force is calledresonance.

Amplitude dependence x m forced oscillations from the frequency ω of the driving force is called resonant characteristic or resonance curve.

Resonance curves at various attenuation levels:

1 – oscillatory system without friction; at resonance, the amplitude x m of forced oscillations increases indefinitely;

2, 3, 4 – real resonance curves for oscillatory systems with different friction.

In the absence of friction, the amplitude of forced oscillations during resonance should increase without limit. In real conditions, the amplitude of steady-state forced oscillations is determined by the condition: the work of the external force during the oscillation period must be equal to the loss of mechanical energy during the same time due to friction. The less friction, the greater the amplitude of forced oscillations during resonance.