A spring pendulum is a material point of mass , attached to an absolutely elastic weightless spring with stiffness . There are two simplest cases: horizontal (Fig. 15, A) and vertical (Fig. 15, b) pendulums.

A) Horizontal pendulum(Fig. 15a). When shifting cargo
out of equilibrium by the amount acts on it in a horizontal direction. restoring elastic force
(Hooke's law).

It is assumed that the horizontal support on which the load slides
during its vibrations, it is absolutely smooth (no friction).

b) vertical pendulum(fig.15, b). The equilibrium position in this case is characterized by the condition:

Where - the magnitude of the elastic force acting on the load
when the spring is statically stretched under the influence of gravity
.

A

Fig.15. Spring pendulum: A- horizontal and b– vertical

If the spring is stretched and the load is released, it will begin to oscillate vertically. If the offset at some point in time is
, then the elastic force will now be written as
.

In both cases considered, the spring pendulum performs harmonic oscillations with a period

(27)

and cyclic frequency

. (28)

Using the example of considering a spring pendulum, we can conclude that harmonic oscillations are a movement caused by a force that increases in proportion to the displacement . Thus, if the restoring force looks like Hooke's law
(she got the namequasi-elastic force ), then the system must perform harmonic oscillations. At the moment of passing the equilibrium position, the restoring force does not act on the body, however, the body skips the equilibrium position by inertia and the restoring force changes direction to the opposite.

Mathematical pendulum

Fig.16. Mathematical pendulum

Mathematical pendulum is an idealized system in the form of a material point suspended on a weightless inextensible thread of length , which performs small oscillations under the action of gravity (Fig. 16).

Oscillations of such a pendulum at small deflection angles
(not exceeding 5º) can be considered harmonic, and the cyclic frequency of the mathematical pendulum:

, (29)

and the period:

. (30)

2.3. Body energy during harmonic vibrations

The energy imparted to the oscillating system during the initial push will be periodically transformed: the potential energy of the deformed spring will be converted into the kinetic energy of the moving load and vice versa.

Let the spring pendulum perform harmonic oscillations with the initial phase
, i.e.
(fig.17).

Fig.17. Law of conservation of mechanical energy

when the spring pendulum oscillates

At the maximum deviation of the load from the equilibrium position, the total mechanical energy of the pendulum (the energy of a deformed spring with stiffness ) is equal to
. When passing through the equilibrium position (
) the potential energy of the spring will become equal to zero, and the total mechanical energy of the oscillatory system will be determined as
.

Figure 18 shows the dependences of the kinetic, potential and total energy in cases where harmonic oscillations are described by trigonometric functions of the sine (dashed line) or cosine (solid line).

Fig.18. Graphs of the time dependence of the kinetic

and potential energy for harmonic oscillations

From the graphs (Fig. 18) it follows that the frequency of change in kinetic and potential energy is twice as high as the natural frequency of harmonic oscillations.

Bodies under the action of an elastic force, the potential energy of which is proportional to the square of the displacement of the body from the equilibrium position:

where k is the stiffness of the spring.

With free mechanical vibrations, the kinetic and potential energies change periodically. At the maximum deviation of the body from the equilibrium position, its velocity, and hence the kinetic energy, vanishes. In this position, the potential energy of the oscillating body reaches its maximum value. For a load on a horizontally located spring, the potential energy is the energy of elastic deformations of the spring.

When the body in its motion passes through the equilibrium position, its speed is maximum. At this moment, it has the maximum kinetic and minimum potential energy. An increase in kinetic energy occurs at the expense of a decrease in potential energy. With further movement, the potential energy begins to increase due to the 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 vibrations remains unchanged.

For spring load:

The start of the oscillatory movement of the body is carried out using the Start button. The Stop button allows you to stop the process at any time.

Graphically shows the relationship between potential and kinetic energies during oscillations at any time. Note that in the absence of damping, the total energy of the oscillatory system remains unchanged, the potential energy reaches its maximum at the maximum deviation of the body from the equilibrium position, and the kinetic energy reaches its maximum value when the body passes through the equilibrium position.

Oscillations of a massive body due to the action of an elastic force

Animation

Description

When an elastic force acts on a massive body, returning it to the equilibrium position, it oscillates around this position.

Such a body is called a spring pendulum. The vibrations are caused by an external force. Oscillations that continue after the external force has ceased to act are called free oscillations. Oscillations caused by the action of an external force are called forced. In this case, the force itself is called compelling.

In the simplest case, a spring pendulum is a solid body moving along a horizontal plane, attached to a wall by a spring (Fig. 1).

Spring pendulum

Rice. 1

The rectilinear motion of a body is described by the dependence of its coordinates on time:

x = x(t). (1)

If all the forces acting on the body under consideration are known, then this dependence can be established using Newton's second law:

md 2 x /dt 2 = S F , (2)

where m is the mass of the body.

The right side of equation (2) is the sum of the projections onto the x axis of all forces acting on the body.

In the case under consideration, the main role is played by the elastic force, which is conservative and can be represented as:

F (x) \u003d - dU (x) / dx, (3)

where U = U(x) is the potential energy of the deformed spring.

Let x be the extension of the spring. It has been experimentally established that at small values ​​of the relative elongation of the spring, i.e. provided that:

S x S<< l ,

where l is the length of the undeformed spring.

Approximately fair dependence:

U (x) = k x 2 /2, (4)

where the coefficient k is called the stiffness of the spring.

From this formula follows the following expression for the elastic force:

F(x) = -kx. (5)

This relationship is called Hooke's law.

In addition to the elastic force, a friction force can act on a body moving along a plane, which is satisfactorily described by the empirical formula:

F tr \u003d - r dx / dt , (6)

where r is the coefficient of friction.

Taking into account formulas (5) and (6), equation (2) can be written as follows:

md 2 x /dt 2 + rdx /dt + kx = F (t ), (7)

where F(t) is the external force.

If only the Hooke force (5) acts on the body, then the free oscillations of the body will be harmonic. Such a body is called a harmonic spring pendulum.

Newton's second law in this case leads to the equation:

d 2 x /dt 2 + w 0 2 x = 0, (8)

w 0 \u003d sqrt (k / m) (9)

Oscillation frequency.

The general solution of equation (8) has the form:

x (t) \u003d A cos (w 0 t + a), (10)

where the amplitude A and the initial phase a are determined by the initial conditions.

When only the elastic force (5) acts on the body under consideration, its total mechanical energy does not change with time:

mv 2 / 2 + k x 2 /2 = const . (eleven)

This statement is the content of the law of conservation of energy of a harmonic spring pendulum.

Let a massive body, in addition to the elastic force that returns it to the equilibrium position, be affected by the friction force. In this case, the free vibrations of the body excited at some point in time will damp out over time and the body will tend to the equilibrium position.

In this Newton's second law (7) can be written as follows:

m d 2 x /dt 2 + rdx /dt + kx = 0, (12)

where m is the mass of the body.

The general solution of equation (12) has the form:

x(t) = a exp(- b t )cos (w t + a ), (13)

w = sqrt(w o 2 - b 2 ) (14)

Oscillation frequency,

b = r / 2 m (15)

The oscillation damping coefficient, amplitude a, and initial phase a are determined by the initial conditions. Function (13) describes the so-called damped oscillations.

The total mechanical energy of the spring pendulum, i.e. the sum of its kinetic and potential energies

E \u003d m v 2 / 2 + kx 2 / 2 (16)

changes over time according to the law:

dE / dt = P , (17)

where P = - rv 2 is the power of the friction force, i.e. energy converted into heat per unit of time.

Timing

Initiation time (log to -3 to -1);

Lifetime (log tc from 1 to 15);

Degradation time (log td -3 to 3);

Optimal development time (log tk -3 to -2).

The operation of most mechanisms is based on the simplest laws of physics and mathematics. The concept of a spring pendulum has become quite widespread. Such a mechanism has become very widespread, since the spring provides the required functionality, it can be an element of automatic devices. Let us consider in more detail such a device, the principle of operation and many other points in more detail.

Spring pendulum definitions

As previously noted, the spring pendulum has become very widespread. Among the features are the following:

1. The device is represented by a combination of weight and spring, the mass of which may not be taken into account. A variety of objects can act as a load. In this case, it can be influenced by an external force. A common example is the creation of a safety valve that is installed in a pipeline system. The fastening of the load to the spring is carried out in a variety of ways. In this case, only the classic screw version is used, which is most widely used. The main properties largely depend on the type of material used in the manufacture, the diameter of the coil, the correct alignment and many other points. End turns are often made in such a way that they can take a large load during operation.
2. Before the deformation begins, the total mechanical energy is absent. In this case, the body is not affected by the force of elasticity. Each spring has its original position, which it maintains for a long period. However, due to a certain rigidity, the body is fixed in its initial position. What matters is how the effort is applied. An example is that it should be directed along the axis of the spring, since otherwise there is a possibility of deformation and many other problems. Each spring has its own specific compression and extension limits. In this case, the maximum compression is represented by the absence of a gap between the individual turns; during tension, there is a moment when an irreversible deformation of the product occurs. If the wire is elongated too much, a change in the basic properties occurs, after which the product does not return to its original position.
3. In the case under consideration, the oscillations are performed due to the action of the elastic force. It is characterized by a fairly large number of features that must be taken into account. The impact of elasticity is achieved due to the specific arrangement of the turns and the type of material used in the manufacture. In this case, the elastic force can act in both directions. Most often, compression occurs, but tension can also be performed - it all depends on the characteristics of the particular case.
4. The speed of movement of the body can vary in a fairly large range, it all depends on what kind of impact. For example, a spring pendulum can move a suspended load in a horizontal and vertical plane. The action of the directional force largely depends on the vertical or horizontal installation.

In general, we can say that the definition of a spring pendulum is rather generalized. In this case, the speed of movement of an object depends on various parameters, for example, the magnitude of the applied force and other moments. Before the actual calculations, a scheme is created:

1. The support to which the spring is attached is indicated. Often, a line with back hatching is drawn to display it.
2. A spring is shown schematically. It is often represented by a wavy line. With a schematic display, the length and diametrical indicator do not matter.
3. The body is also depicted. It should not correspond to the dimensions, however, the place of direct attachment matters.

The diagram is required to schematically display all the forces that affect the device. Only in this case it is possible to take into account everything that affects the speed of movement, inertia and many other moments.

Spring pendulums are used not only in calculations or solving various problems, but also in practice. However, not all properties of such a mechanism are applicable.

An example is the case when oscillatory movements are not required:

1. Creation of locking elements.
2. Spring mechanisms associated with the transportation of various materials and objects.

Conducted calculations of the spring pendulum allow you to choose the most suitable body weight, as well as the type of spring. It is characterized by the following features:

1. Winding diameter. It can be very different. How much material is required for production largely depends on the diameter indicator. The diameter of the coils also determines how much force must be applied to fully compress or partially expand. However, an increase in size can create significant difficulties in installing the product.
2. The diameter of the wire. Another important parameter is the diameter of the wire. It can vary over a wide range, depending on the strength and degree of elasticity.
3. Length of the product. This indicator determines how much force is required for full compression, as well as how much elasticity the product can have.
4. The type of material used also determines the basic properties. Most often, the spring is made using a special alloy that has the appropriate properties.

In mathematical calculations, many points are not taken into account. Elastic force and many other indicators are determined by calculation.

Types of spring pendulum

There are several different types of spring pendulum. It should be borne in mind that classification can be carried out according to the type of spring being installed. Among the features we note:

1. Quite widespread are vertical oscillations, since in this case the load does not have friction and other effects. With a vertical arrangement of the load, the degree of influence of gravity increases significantly. This variant of execution is widespread when carrying out a variety of calculations. Due to gravity, it is likely that the body at the starting point will make a large number of inertial movements. This is also facilitated by the elasticity and inertia of the movement of the body at the end of the stroke.
2. A horizontal spring pendulum is also used. In this case, the load is on the supporting surface and friction also occurs at the moment of movement. When placed horizontally, gravity works a little differently. The horizontal position of the body has become widespread in various tasks.

The movement of a spring pendulum can be calculated using a sufficiently large number of different formulas, which must take into account the impact of all forces. In most cases, a classic spring is installed. Among the features we note the following:

1. The classic twisted compression spring is very widespread today. In this case, there is a space between the turns, which is called a pitch. The compression spring can be stretched, but often it is not installed for this. A distinctive feature can be called the fact that the last turns are made in the form of a plane, due to which a uniform distribution of force is ensured.
2. A stretch version can be installed. It is designed to be installed when the applied force causes an increase in length. Hooks are placed for fastening.

This results in an oscillation that can last for a long period. The above formula allows you to calculate taking into account all the moments.

Formulas for the period and frequency of oscillation of a spring pendulum

When designing and calculating key indicators, quite a lot of attention is also paid to the frequency and period of oscillation. Cosine is a periodic function that uses a value that does not change after a certain period of time. It is this indicator that is called the period of oscillation of a spring pendulum. The letter T is used to designate this indicator, and the concept is often used to characterize the value inverse to the period of oscillation (v). In most cases, the formula T=1/v is used in calculations.

The oscillation period is calculated using a somewhat complicated formula. It is as follows: T=2p√m/k. To determine the oscillation frequency, the formula is used: v=1/2п√k/m.

Considered cyclic oscillation frequency of the spring pendulum depends on the following points:

1. The mass of the weight that is attached to the spring. This indicator is considered the most important, as it affects a variety of parameters. The force of inertia, speed and many other indicators depend on the mass. In addition, the mass of the load is a quantity that is not difficult to measure due to the presence of special measuring equipment.
2. elasticity coefficient. For each spring, this indicator is significantly different. The coefficient of elasticity is indicated to determine the main parameters of the spring. This parameter depends on the number of turns, the length of the product, the distance between the turns, their diameter and much more. It is determined in a variety of ways, often with the use of special equipment.

Do not forget that when the spring is strongly stretched, Hooke's law ceases to operate. In this case, the period of the spring oscillation begins to depend on the amplitude.

The period is measured in the universal unit of time, in most cases seconds. In most cases, the oscillation amplitude is calculated when solving a variety of problems. To simplify the process, a simplified diagram is constructed, which displays the main forces.

Formulas for the amplitude and initial phase of a spring pendulum

Having decided on the features of the processes being passed and knowing the equation of oscillations of the spring pendulum, as well as the initial values, it is possible to calculate the amplitude and initial phase of the spring pendulum. The value of f is used to determine the initial phase, the amplitude is denoted by the symbol A.

To determine the amplitude, the formula can be used: A \u003d √x 2 + v 2 / w 2. The initial phase is calculated by the formula: tgf=-v/xw.

Using these formulas, it is possible to determine the main parameters that are used in the calculations.

Energy of oscillations of a spring pendulum

When considering the oscillation of a load on a spring, one must take into account the moment that the movement of the pendulum can be described by two points, that is, it is rectilinear. This moment determines the fulfillment of the conditions concerning the force in question. We can say that the total energy is potential.

It is possible to calculate the energy of oscillations of a spring pendulum, taking into account all the features. Let's name the following as the main points:

1. Oscillations can take place in the horizontal and vertical plane.
2. Zero potential energy is chosen as the equilibrium position. This is where the origin of coordinates is set. As a rule, in this position, the spring retains its shape, provided that there is no deforming force.
3. In the case under consideration, the calculated energy of the spring pendulum does not take into account the friction force. With a vertical load, the friction force is insignificant, with a horizontal load, the body is on the surface and friction may occur during movement.
4. The following formula is used to calculate the vibration energy: E=-dF/dx.

The above information indicates that the law of conservation of energy is as follows: mx 2 /2+mw 2 x 2 /2=const. The applied formula says the following:

It is possible to determine the energy of oscillation of a spring pendulum when solving a variety of problems.

Free oscillations of a spring pendulum

Considering what caused the free oscillations of a spring pendulum, attention should be paid to the action of internal forces. They begin to form almost immediately after the movement has been transferred to the body. Features of harmonic oscillations are in the following points:

1. Other types of forces of an influencing nature may also arise, which satisfy all the norms of the law, are called quasi-elastic.
2. The main reasons for the operation of the law may be internal forces that are formed immediately at the moment of changing the position of the body in space. In this case, the load has a certain mass, the force is created by fixing one end for a stationary object with sufficient strength, the second for the load itself. In the absence of friction, the body can perform oscillatory motions. In this case, the fixed load is called linear.

Do not forget that there is simply a huge number of different types of systems in which the movement of an oscillatory nature is carried out. Elastic deformation also occurs in them, which causes them to be used to perform any work.