Bank of ready-made tasks. Dynamics of oscillatory motion – Knowledge Hypermarket

LECTURE No. 8

Mechanics

Oscillations

Oscillatory movement. Kinematic and dynamic characteristics of oscillatory motion. Mathematical, physical and spring pendulum.

We live in a world where oscillatory processes are an integral part of our world and are found everywhere.

An oscillatory process or oscillation is a process characterized by varying degrees of repeatability.

If a oscillating quantity repeats its values ​​at equal intervals of time, then such oscillations are called periodic, and these time intervals are called the period of oscillation.

Depending on the physical nature of the phenomenon, vibrations are distinguished: mechanical, electromechanical, electromagnetic, etc.

Oscillations are widespread in nature and technology. Oscillatory processes underlie some branches of mechanics. In this course of lectures we will only talk about mechanical vibrations.

Depending on the nature of the impact on the oscillatory system, vibrations are distinguished: 1. Free or natural, 2. Forced vibrations, 3. Self-oscillations, 4. Parametric vibrations.

Free vibrations are vibrations that occur without external influence and are caused by an initial “push”.

Forced oscillations occur under the influence of a periodic external force

Self-oscillations also occur under the influence of an external force, but the moment of influence of the force on the system is determined by the oscillatory system itself.

With parametric oscillations, due to external influences, a periodic change in the parameters of the system occurs, which causes this type of oscillation.

The simplest form is harmonic vibrations

Harmonic oscillations are vibrations that occur according to the lawsin orcos . An example of harmonic oscillations is the oscillation of a mathematical pendulum

The maximum deviation of an oscillating quantity during the oscillation process is called amplitude of oscillations(A) . The time it takes to complete one complete oscillation is called period of oscillation(T) . The reciprocal of the oscillation period is called vibration frequency(). Often vibrations multiplied by 2 are called cyclic frequency(). Thus, harmonic vibrations are described by the expression

Here ( t+  0 ) oscillation phase, and  0 – initial phase

The simplest mechanical oscillatory systems are the so-called mathematical, spring and physical pendulums. Let's look at these pendulums in more detail

8.1. Math pendulum

A mathematical pendulum is an oscillatory system consisting of a massive point body suspended in a field of gravity on an inextensible weightless thread.

At the bottom point the pendulum has a minimum of potential energy. Let's deflect the pendulum by an angle . The center of gravity of a massive point body will rise to a height  h and at the same time the potential energy of the pendulum will increase by the amount mg h. In addition, in a deflected position, the load is affected by gravity and the tension of the thread. The lines of action of these forces do not coincide, and a resultant force acts on the load, tending to return it to the equilibrium position. If the load is not held, then under the influence of this force it will begin to move to its original equilibrium position, its kinetic energy will increase due to an increase in speed, while the potential energy will decrease. When the equilibrium point is reached, the resulting force will no longer act on the body (the force of gravity at this point is compensated by the tension force of the thread). The potential energy of the body at this point will be minimal, and the kinetic energy, on the contrary, will have its maximum value. The body, moving by inertia, will pass the equilibrium position and begin to move away from it, which will lead to the emergence of a resultant force (from the force of tension and gravity), which will be directed against the movement of the body, braking it. At the same time, the kinetic energy of the load begins to decrease and its potential energy increases. This process will continue until the reserves of kinetic energy are completely exhausted and converted into potential energy. In this case, the deviation of the load from the equilibrium position will reach its maximum value and the process will repeat. If there is no friction in the system, the load will oscillate indefinitely.

Thus, oscillatory mechanical systems are characterized by the fact that when they deviate from the equilibrium position, a restoring force arises in the system, tending to return the system to the equilibrium position. In this case, oscillations occur, accompanied by a periodic transition of the potential energy of the system into its kinetic energy and vice versa.

Let's calculate the oscillatory process. moment of force M acting on the pendulum is obviously equal to - mglsin The minus sign reflects the fact that the moment of force tends to return the load to the equilibrium position. On the other hand, according to the basic law of rotational motion M=ID 2  / dt 2 . Thus, we obtain the equality

B
We will consider only small angles of deviation of the pendulum from the equilibrium position. Then sin ≈  . And our equality will take the form:

D
For a mathematical pendulum it is true I= ml 2 . Substituting this equality into the resulting expression, we obtain an equation describing the process of oscillation of a mathematical pendulum:

This differential equation describes the oscillatory process. The solution to this equation is the harmonic functions sin( t+  0 ) or cos ( t+  0 ) Indeed, we substitute any of these functions into the equation and get:  2 = g/ l. Thus, if this condition is met, then the functions sin( t+  0 ) or cos( t+  0 ) transform the differential equation of oscillations into an identity.

ABOUT
Here the cyclic frequency and period of oscillation of a harmonic pendulum is expressed as:

The amplitude of oscillations is found from the initial conditions of the problem.

As we can see, the frequency and period of oscillation of a mathematical pendulum does not depend on the mass of the load and depends only on the acceleration of free fall and the length of the suspension thread, which allows the pendulum to be used as a simple but very accurate device for determining the acceleration of free fall.

Another type of pendulum is any physical body suspended from some point of the body and having the ability to perform an oscillatory movement.

8.2. Physical pendulum

IN Let's take an arbitrary body, pierce it at some point with an axis that does not coincide with its center of mass, around which the body can freely rotate. Let us suspend the body on this axis and deflect it from the equilibrium position by a certain angle  .

T
when on a body with a moment of inertia I relative to the axis ABOUT there will be a moment returning to the equilibrium position M = - mglsin and the oscillations of a physical pendulum, like a mathematical one, will be described by the differential equation:

Since for different physical pendulums the moment of inertia will be expressed differently, we will not describe it as in the case of a mathematical pendulum. This equation also has the form of an oscillation equation, the solution of which is the functions describing harmonic oscillations. In this case, the cyclic frequency ( ) , oscillation period (T) are defined as:

We see that in the case of a physical pendulum, the period of oscillation depends on the geometry of the pendulum body, and not on its mass, as in the case of a mathematical pendulum. Indeed, the expression for the moment of inertia includes the mass of the pendulum to the first power. The moment of inertia in the expression for the period of oscillation is in the numerator, while the mass of the pendulum is in the denominator and also to the first power. Thus, the mass in the numerator cancels with the mass in the denominator.

A physical pendulum has one more characteristic: reduced length.

The reduced length of a physical pendulum is the length of a mathematical pendulum, the period of which coincides with the period of the physical pendulum.

This definition makes it easy to define an expression for the given length.

Comparing these expressions we get

If on a line drawn from the point of suspension through the center of mass of the physical pendulum we plot (starting from the point of suspension) the reduced length of the physical pendulum, then at the end of this segment there will be a point that has a remarkable property. If a physical pendulum is suspended from this point, then its period of oscillation will be the same as in the case of hanging the pendulum at the previous suspension point. These points are called the swing centers of the physical pendulum.

Let's consider another simple oscillatory system that performs harmonic oscillations

8.3. Spring pendulum

P Let us imagine that at the end of a spring with a stiffness coefficient k attached mass m.

If we move the load along the x-axis by stretching the spring, then a force returning to the equilibrium position will act on the load F return = - kx. If the load is released, this force will cause acceleration d 2 x / dt 2 . According to Newton's second law we get:

md 2 x / dt 2 = - kx from this equation we obtain the equation for the oscillation of a load on a spring in its final form: d 2 x / dt 2 + (k/ m) x = 0

E
then the equation of oscillations has the same form as the equations of oscillations in the cases already considered, which means that the solution to this equation will be the same harmonic functions. The frequency and period of oscillations will be respectively equal

Moreover, gravity in no way affects the oscillations of the spring pendulum. Since in this case it is a constantly acting factor, acting all the time in one direction and having nothing to do with the restoring force.

Thus, as we see the oscillatory process in a mechanical oscillatory system, it is characterized primarily by the presence in the system restoring force acting on the system, and the oscillations themselves are characterized by: amplitude of oscillations, their period, frequency and phase of oscillations.

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lungs

heart

Lesson topic: “Free and forced vibrations. Dynamics of oscillatory motion".

• Mechanical vibrations – these are movements that are repeated exactly or approximately at certain intervals of time.

Main types of vibrations

forced

free

called vibrations of bodies under the influence of external periodically changing forces.

are called oscillations in a system under the influence of internal forces, after the system has been removed from an equilibrium position and then left to itself.

Pendulum - a body suspended on a thread or fixed to an axis that can oscillate under the influence of gravity

Types of pendulums

Spring- a body suspended on a spring and oscillating under the action of the elastic force of the spring.

Mathematical (thread) is a material point suspended on a weightless and inextensible thread.

Conditions for the occurrence of oscillations

• When a body is removed from an equilibrium position, a force arises in the system, directed towards the equilibrium position and, therefore, tending to return the body to the equilibrium position.

• The friction in the system should be quite low.

• Amplitude – modulus of the greatest displacement of the body from the equilibrium position.

X max or A

Measured in meters

• Period T – the time of one complete oscillation.

Measured in seconds

Oscillation period

For math

pendulum

For spring

pendulum

(Huygens formula)

Frequency - the number of complete oscillations per unit time.

Measured in Hertz

• Cyclic (circular) vibration frequency – frequency equal to the number of oscillations performed by a material point per

Measured in radians per second

World of fluctuations

• Oscillations are one of the most common processes in nature and technology.
• wings of insects and birds in flight,
• high-rise buildings and high-voltage wires exposed to wind,
• pendulum of a wound clock and a car on springs while driving
• river level throughout the year and human body temperature during illness.

A little history...

Galileo Galilei (1564-1642)

The great Italian scientist is one of the creators of exact natural science.

One day in church he I watched the huge chandelier swing and timed the time by my pulse. He later discovered that the time it takes to swing once depends on the length of the pendulum - the time is reduced by half if the pendulum is shortened by three quarters.

A little history...

The most famous practical use of a pendulum is its use in clocks to measure time. This was first done by the Dutch physicist H. Huygens. The scientist worked on the task of creating and improving clocks, primarily pendulum ones, for almost forty years: from 1656 to 1693, Huygens derived a formula for determining the period of oscillation of a mathematical pendulum. Before this, time was measured by the flow of water, the burning of a torch or candle.

Foucault pendulum

In 1850, J. Foucault suspended a pendulum under the dome of a tall building so that the tip of the pendulum, when swinging, left a mark on the sand poured on the floor. It turned out that with each roll the tip leaves a new mark in the sand.

Thus, Foucault's experiment showed that the Earth rotates around its axis.

At first, the experiment was carried out in a narrow circle, but Napoleon was so interested III, the French Emperor, that he suggested to Foucault that it be repeated publicly on a grand scale under the dome of the Pantheon in Paris. This public demonstration is usually called the Foucault experiment.

In geology, a pendulum is used to experimentally determine a numerical value g at different points on the earth's surface. To do this, based on a sufficiently large number of oscillations of the pendulum in the place where they measure g , find the period of its oscillations T, and g calculated using the formula:

Noticeable deviation in value g from the norm for any area is called a gravitational anomaly. Anomaly detection helps locate mineral deposits.

Laboratory work "Determination of the acceleration of free fall using a pendulum"

Goal of the work: Learn experimentally to measure the acceleration of free fall using a mathematical pendulum.

Equipment: tripod, ball on a string, clock, ruler.

From the three proposed verses, choose one that characterizes your state at the end of the lesson .

1.Eyes sparkle The soul laughs And my mind sings: “Forward to knowledge”!

2. I’m not happy today In the silence I felt sad, All about fluctuations flashed in the distance.

3. Remembering all your knowledge, And physicists comprehend the world, I am grateful to mother fate, That there are fluctuations in the world

and we can’t count them all!

A mathematical pendulum is a model of an ordinary pendulum. A mathematical pendulum is a material point suspended on a long weightless and inextensible thread.

Let's move the ball out of its equilibrium position and release it. Two forces will act on the ball: gravity and the tension of the thread. When the pendulum moves, the force of air friction will still act on it. But we will consider it very small.

Let us decompose the force of gravity into two components: a force directed along the thread, and a force directed perpendicular to the tangent to the trajectory of the ball.

These two forces add up to the force of gravity. The elastic forces of the thread and the gravity component Fn impart centripetal acceleration to the ball. The work done by these forces will be zero, and therefore they will only change the direction of the velocity vector. At any moment in time, it will be directed tangentially to the arc of the circle.

Under the influence of the gravity component Fτ, the ball will move along a circular arc with a speed increasing in magnitude. The value of this force always changes in magnitude; when passing through the equilibrium position, it is equal to zero.

Dynamics of oscillatory motion

Equation of motion of a body oscillating under the action of an elastic force.

General equation of motion:

Vibrations in the system occur under the influence of elastic force, which, according to Hooke's law, is directly proportional to the displacement of the load

Then the equation of motion of the ball will take the following form:

Divide this equation by m, we get the following formula:

And since the mass and elasticity coefficient are constant values, then the ratio (-k/m) will also be constant. We have obtained an equation that describes the vibrations of a body under the action of elastic force.

The projection of the acceleration of the body will be directly proportional to its coordinate, taken with the opposite sign.

Equation of motion of a mathematical pendulum

The equation of motion of a mathematical pendulum is described by the following formula:

This equation has the same form as the equation of motion of a mass on a spring. Consequently, the oscillations of the pendulum and the movements of the ball on the spring occur in the same way.

The displacement of the ball on the spring and the displacement of the pendulum body from the equilibrium position change over time according to the same laws.

In order to quantitatively describe the vibrations of a body under the action of the elastic force of a spring or the vibrations of a ball suspended on a thread, we will use Newton’s laws of mechanics.

Equation of motion of a body oscillating under the action of an elastic force. According to Newton’s second law, the product of a body’s mass m and its acceleration is equal to the resultant F of all forces applied to the body:

This is the equation of motion. Let us write the equation of motion for a ball moving rectilinearly along the horizontal under the action of the elastic force of a spring (see Fig. 3.3). Let's direct the OX axis to the right. Let the origin of the coordinates correspond to the equilibrium position of the ball (see Fig. 3.3, a).

In the projection onto the OX axis, the equation of motion (3.1) can be written as follows: ma x = F x control, where a x and F x control are respectively the projections of the acceleration and elastic force of the spring on this axis.

According to Hooke's law, the projection F x ynp is directly proportional to the displacement of the ball from its equilibrium position. The displacement is equal to the x coordinate of the ball, and the projection of the force and the coordinate have opposite signs (see Fig. 3.3, b, c). Hence,

F x yпp = -khх, (3.2)

Dividing the left and right sides of equation (3.3) by m, we obtain

Since mass m and stiffness k are constant values, their ratio is also a constant value.

We have obtained an equation that describes the vibrations of a body under the action of elastic force. It's very simple: the projection ax of the acceleration of a body is directly proportional to its coordinate x, taken with the opposite sign.

Equation of motion of a mathematical pendulum. When a ball oscillates on an inextensible thread, it constantly moves along an arc of a circle, the radius of which is equal to the length of the thread l. Therefore, the position of the ball at any time is determined by one value - the angle α of the thread’s deviation from the vertical. We will consider the angle α positive if the pendulum is tilted to the right from the equilibrium position, and negative if it is tilted to the left (see Fig. 3.5). The tangent to the trajectory will be considered directed towards the positive angle reference.

Let us denote the projection of gravity onto the tangent to the trajectory of the pendulum by F τ. This projection at the moment when the pendulum thread is deflected from the equilibrium position by an angle α is equal to:

F τ = -mg sin α. (3.5)

The “-” sign is here because the quantities F τ and a have opposite signs. When the pendulum is deflected to the right (α > 0), the gravity component τ is directed to the left and its projection is negative: F τ< 0. При отклонении маятника влево (α < 0) эта проекция положительна: F τ > 0.

Let us denote the projection of the acceleration of the pendulum onto the tangent to its trajectory by a τ. This projection characterizes the speed of change of the pendulum velocity modulus.

According to Newton's second law

ma τ = -mg sin α. (3.6)

Dividing the left and right sides of this equation by m, we get

and τ = -g sin α. (3.7)

Previously it was assumed that the angles of deflection of the pendulum thread from the vertical could be any. In what follows we will consider them small. For small angles, if the angle is measured in radians,

Therefore, one can accept

and τ = -gα. (3.8)

If the angle α is small, then the projection of acceleration is approximately equal to the projection of acceleration on the OX axis: a τ ≈ a x (see Fig. 3.5). From triangle ABO for small angle a we have:

Substituting this expression into equality (3.8) instead of angle α, we obtain

This equation has the same form as equation (3.4) for the acceleration of a ball attached to a spring. Consequently, the solution to this equation will have the same form as the solution to equation (3.4). This means that the movement of the ball and the oscillations of the pendulum occur in the same way. The displacements of the ball on the spring and the pendulum body from the equilibrium positions change over time according to the same law, despite the fact that the forces causing the oscillations have a different physical nature. By multiplying equations (3.4) and (3.10) by m and remembering Newton’s second law mа x = F x peз, we can conclude that oscillations in these two cases occur under the influence of forces, the resultant of which is directly proportional to the displacement of the oscillating body from the equilibrium position and is directed in the direction opposite to this displacement.

Equation (3.4), like (3.10), is apparently very simple: acceleration is directly proportional to the coordinate (displacement from the equilibrium position).