A spring pendulum is a material point with mass attached to an absolutely elastic weightless spring with a stiffness 
. There are two simplest cases: horizontal (Fig. 15, A) and vertical (Fig. 15, b) pendulums.
A)
Horizontal pendulum(Fig. 15, a). When the load moves 
from the equilibrium position 
by the amount 
acts on it in the horizontal direction restoring elastic force
(Hooke's law).
It is assumed that the horizontal support along 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 by 
under the influence of gravity of the load 
.
|
A |
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Fig. 15. Spring pendulum: A– horizontal and b– vertical
If you stretch the spring and release the load, it will begin to oscillate vertically. If the displacement 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 a spring pendulum, we can conclude that harmonic oscillations are motion caused by a force that increases in proportion to the displacement 
. Thus, if the restoring force resembles Hooke's law
(she got the namequasi-elastic force
), then the system must perform harmonic oscillations. At the moment of passing the equilibrium position, no restoring force acts on the body; however, the body, by inertia, passes the equilibrium position and the restoring force changes direction to the opposite.
Math pendulum
Fig. 16.
, which makes small oscillations under the influence of gravity (Fig. 16).
Oscillations of such a pendulum at small angles of deflection 
(not exceeding 5º) can be considered harmonic, and the cyclic frequency of a mathematical pendulum:
,
(29)
and period:
.
(30)
2.3. Body energy during harmonic oscillations
The energy imparted to the oscillatory system during the initial push will be periodically transformed: the potential energy of the deformed spring will transform into the kinetic energy of the moving load and back.
Let the spring pendulum perform harmonic oscillations with the initial phase 
, i.e. 
(Fig. 17).
Fig. 17. Law of conservation of mechanical energy
when a 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 a stiffness 
) is equal to 
. 
When passing 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 graphs of the dependences of kinetic, potential and total energy in cases where harmonic vibrations are described by trigonometric functions of sine (dashed line) or cosine (solid line).
Fig. 18. Graphs of time dependence of kinetic
and potential energy during 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.
10.4. Law of conservation of energy during harmonic oscillations 10.4.1. Energy conservation at
mechanical harmonic vibrations
Conservation of energy during oscillations of a mathematical pendulum
During harmonic vibrations, the total mechanical energy of the system is conserved (remains constant).
Total mechanical energy of a mathematical pendulum
E = W k + W p ,
where W k is kinetic energy, W k = = mv 2 /2; W p - potential energy, W p = mgh; m is the mass of the load; g - free fall acceleration module; v - load speed module; h is the height of the load above the equilibrium position (Fig. 10.15).
During harmonic oscillations, a mathematical pendulum goes through a number of successive states, so it is advisable to consider the energy of a mathematical pendulum in three positions (see Fig. 10.15):
Rice. 10.15 1) in
potential energy is zero; The total energy coincides with the maximum kinetic energy:
E = W k max ;
2) in emergency situation(2) the body is raised above the initial level to a maximum height h max, therefore the potential energy is also maximum:
W p max = m g h max ;
kinetic energy is zero; total energy coincides with maximum potential energy:
E = W p max ;
3) in intermediate position(3) the body has an instantaneous speed v and is raised above the initial level to a certain height h, therefore the total energy is the sum
E = m v 2 2 + m g h ,
where mv 2 /2 is kinetic energy; mgh - potential energy; m is the mass of the load; g - free fall acceleration module; v - load speed module; h is the height of the load above the equilibrium position.
During harmonic oscillations of a mathematical pendulum, the total mechanical energy is conserved:
E = const.
The values of the total energy of the mathematical pendulum in its three positions are reflected in the table. 10.1.
| № | Position | Wp | Wk | E = W p + W k |
|---|---|---|---|---|
| 1 | Equilibrium | 0 | m v max 2 / 2 | m v max 2 / 2 |
| 2 | Extreme | mgh max | 0 | mgh max |
| 3 | Intermediate (instant) | mgh | mv 2 /2 | mv 2 /2 + mgh |
The values of total mechanical energy presented in the last column of the table. 10.1, have equal values for any position of the pendulum, which is a mathematical expression:
m v max 2 2 = m g h max;
m v max 2 2 = m v 2 2 + m g h ;
m g h max = m v 2 2 + m g h ,
where m is the mass of the load; g - free fall acceleration module; v is the module of the instantaneous speed of the load in position 3; h - height of lifting of the load above the equilibrium position in position 3; v max - maximum speed module of the load in position 1; h max - maximum height of lifting the load above the equilibrium position in position 2.
Thread deflection angle mathematical pendulum from the vertical (Fig. 10.15) is determined by the expression
cos α = l − hl = 1 − hl ,
where l is the length of the thread; h is the height of the load above the equilibrium position.
Maximum angle deviation α max is determined by the maximum height of lifting the load above the equilibrium position h max:
cos α max = 1 − h max l .
Example 11. The period of small oscillations of a mathematical pendulum is 0.9 s. What is the maximum angle at which the thread will deviate from the vertical if, passing the equilibrium position, the ball moves at a speed of 1.5 m/s? There is no friction in the system.
Solution . The figure shows two positions of the mathematical pendulum:
- equilibrium position 1 (characterized by the maximum speed of the ball v max);
- extreme position 2 (characterized by the maximum lifting height of the ball h max above the equilibrium position).
The required angle is determined by the equality
cos α max = l − h max l = 1 − h max l ,
where l is the length of the pendulum thread.
We find the maximum height of the pendulum ball above the equilibrium position from the law of conservation of total mechanical energy.
The total energy of the pendulum in the equilibrium position and in the extreme position is determined by the following formulas:
- in a position of balance -
E 1 = m v max 2 2,
where m is the mass of the pendulum ball; v max - module of the ball velocity in the equilibrium position (maximum speed), v max = 1.5 m/s;
- in extreme position -
E 2 = mgh max,
where g is the gravitational acceleration module; h max is the maximum height of the ball lifting above the equilibrium position.
Law of conservation of total mechanical energy:
m v max 2 2 = m g h max .
Let us express from here the maximum height of the ball's rise above the equilibrium position:
h max = v max 2 2 g .
We determine the length of the thread from the formula for the oscillation period of a mathematical pendulum
T = 2 π l g ,
those. thread length
l = T 2 g 4 π 2 .
Let's substitute h max and l into the expression for the cosine of the desired angle:
cos α max = 1 − 2 π 2 v max 2 g 2 T 2
and perform the calculation taking into account the approximate equality π 2 = 10:
cos α max = 1 − 2 ⋅ 10 ⋅ (1.5) 2 10 2 ⋅ (0.9) 2 = 0.5 .
It follows that the maximum deflection angle is 60°.
Strictly speaking, at an angle of 60° the oscillations of the ball are not small and it is unlawful to use the standard formula for the period of oscillation of a mathematical pendulum.
Conservation of energy during oscillations of a spring pendulum
Total mechanical energy of a spring pendulum consists of kinetic energy and potential energy:
Total mechanical energy of a mathematical pendulum
where W k is kinetic energy, W k = mv 2 /2; W p - potential energy, W p = k (Δx ) 2 /2; m is the mass of the load; v - load speed module; k is the stiffness (elasticity) coefficient of the spring; Δx - deformation (tension or compression) of the spring (Fig. 10.16).
In the International System of Units, the energy of a mechanical oscillatory system is measured in joules (1 J).
During harmonic oscillations, the spring pendulum goes through a number of successive states, so it is advisable to consider the energy of the spring pendulum in three positions (see Fig. 10.16):
Rice. 10.15 1) in(1) the speed of the body has a maximum value v max, therefore the kinetic energy is also maximum:
W k max = m v max 2 2 ;
the potential energy of the spring is zero, since the spring is not deformed; The total energy coincides with the maximum kinetic energy:
E = W k max ;
2) in emergency situation(2) the spring has a maximum deformation (Δx max), so the potential energy also has a maximum value:
W p max = k (Δ x max) 2 2 ;
the kinetic energy of the body is zero; total energy coincides with maximum potential energy:
E = W p max ;
3) in intermediate position(3) the body has an instantaneous speed v, the spring has some deformation at this moment (Δx), so the total energy is the sum
E = m v 2 2 + k (Δ x) 2 2 ,
where mv 2 /2 is kinetic energy; k (Δx) 2 /2 - potential energy; m is the mass of the load; v - load speed module; k is the stiffness (elasticity) coefficient of the spring; Δx - deformation (tension or compression) of the spring.
When the load of a spring pendulum is displaced from its equilibrium position, it is acted upon by restoring force, the projection of which onto the direction of movement of the pendulum is determined by the formula
F x = −kx ,
where x is the displacement of the spring pendulum load from the equilibrium position, x = ∆x, ∆x is the deformation of the spring; k is the stiffness (elasticity) coefficient of the pendulum spring.
During harmonic oscillations of a spring pendulum, the total mechanical energy is conserved:
E = const.
The values of the total energy of the spring pendulum in its three positions are reflected in the table. 10.2.
| № | Position | Wp | Wk | E = W p + W k |
|---|---|---|---|---|
| 1 | Equilibrium | 0 | m v max 2 / 2 | m v max 2 / 2 |
| 2 | Extreme | k (Δx max) 2 /2 | 0 | k (Δx max) 2 /2 |
| 3 | Intermediate (instant) | k (Δx ) 2 /2 | mv 2 /2 | mv 2 /2 + k (Δx ) 2 /2 |
The values of total mechanical energy presented in the last column of the table have equal values for any position of the pendulum, which is a mathematical expression law of conservation of total mechanical energy:
m v max 2 2 = k (Δ x max) 2 2 ;
m v max 2 2 = m v 2 2 + k (Δ x) 2 2 ;
k (Δ x max) 2 2 = m v 2 2 + k (Δ x) 2 2 ,
where m is the mass of the load; v is the module of the instantaneous speed of the load in position 3; Δx - deformation (tension or compression) of the spring in position 3; v max - maximum speed module of the load in position 1; Δx max - maximum deformation (tension or compression) of the spring in position 2.
Example 12. A spring pendulum performs harmonic oscillations. How many times is its kinetic energy greater than its potential energy at the moment when the displacement of the body from the equilibrium position is a quarter of the amplitude?
Solution . Let's compare two positions of the spring pendulum:
- extreme position 1 (characterized by the maximum displacement of the pendulum load from the equilibrium position x max);
- intermediate position 2 (characterized by intermediate values of displacement from the equilibrium position x and velocity v →).
The total energy of the pendulum in the extreme and intermediate positions is determined by the following formulas:
- in extreme position -
E 1 = k (Δ x max) 2 2,
where k is the stiffness (elasticity) coefficient of the spring; ∆x max - amplitude of oscillations (maximum displacement from the equilibrium position), ∆x max = A;
- in an intermediate position -
E 2 = k (Δ x) 2 2 + m v 2 2 ,
where m is the mass of the pendulum load; ∆x - displacement of the load from the equilibrium position, ∆x = A /4.
The law of conservation of total mechanical energy for a spring pendulum has the following form:
k (Δ x max) 2 2 = k (Δ x) 2 2 + m v 2 2 .
Let us divide both sides of the written equality by k (∆x) 2 /2:
(Δ x max Δ x) 2 = 1 + m v 2 2 ⋅ 2 k Δ x 2 = 1 + W k W p ,
where W k is the kinetic energy of the pendulum in an intermediate position, W k = mv 2 /2; W p - potential energy of the pendulum in an intermediate position, W p = k (∆x ) 2 /2.
Let us express the required energy ratio from the equation:
W k W p = (Δ x max Δ x) 2 − 1
and calculate its value:
W k W p = (A A / 4) 2 − 1 = 16 − 1 = 15 .
At the indicated moment of time, the ratio of the kinetic and potential energies of the pendulum is 15.
Definition
Oscillation frequency($\nu$) is one of the parameters that characterize oscillations. This is the reciprocal of the oscillation period ($T$):
\[\nu =\frac(1)(T)\left(1\right).\]
Thus, the oscillation frequency is a physical quantity equal to the number of repetitions of oscillations per unit time.
\[\nu =\frac(N)(\Delta t)\left(2\right),\]
where $N$ is the number of complete oscillatory movements; $\Delta t$ is the time during which these oscillations occurred.
The cyclic oscillation frequency ($(\omega )_0$) is related to the frequency $\nu $ by the formula:
\[\nu =\frac((\omega )_0)(2\pi )\left(3\right).\]
The unit of frequency in the International System of Units (SI) is the hertz or reciprocal second:
\[\left[\nu \right]=с^(-1)=Hz.\]
Spring pendulum
Definition
Spring pendulum called a system that consists of an elastic spring to which a load is attached.
Let us assume that the mass of the load is $m$ and the elasticity coefficient of the spring is $k$. The mass of the spring in such a pendulum is usually not taken into account. If we consider the horizontal movements of the load (Fig. 1), then it moves under the influence of elastic force if the system is taken out of equilibrium and left to its own devices. In this case, it is often believed that friction forces can be ignored.
Equations of oscillations of a spring pendulum
A spring pendulum that oscillates freely is an example of a harmonic oscillator. Let him oscillate along the X axis. If the oscillations are small, Hooke’s law is satisfied, then we write the equation of motion of the load as:
\[\ddot(x)+(\omega )^2_0x=0\left(4\right),\]
where $(\omega )^2_0=\frac(k)(m)$ is the cyclic frequency of oscillations of the spring pendulum. The solution to equation (4) is a sine or cosine function of the form:
where $(\omega )_0=\sqrt(\frac(k)(m))>0$ is the cyclic frequency of oscillations of the spring pendulum, $A$ is the amplitude of oscillations; $((\omega )_0t+\varphi)$ - oscillation phase; $\varphi $ and $(\varphi )_1$ are the initial phases of oscillations.
Oscillation frequency of a spring pendulum
From formula (3) and $(\omega )_0=\sqrt(\frac(k)(m))$, it follows that the oscillation frequency of the spring pendulum is equal to:
\[\nu =\frac(1)(2\pi )\sqrt(\frac(k)(m))\ \left(6\right).\]
Formula (6) is valid if:
- the spring in the pendulum is considered weightless;
- the load attached to the spring is an absolutely rigid body;
- there are no torsional vibrations.
Expression (6) shows that the oscillation frequency of the spring pendulum increases with decreasing mass of the load and increasing the elasticity coefficient of the spring. The oscillation frequency of a spring pendulum does not depend on the amplitude. If the oscillations are not small, the elastic force of the spring does not obey Hooke's law, then a dependence of the oscillation frequency on the amplitude appears.
Examples of problems with solutions
Example 1
Exercise. The period of oscillation of a spring pendulum is $T=5\cdot (10)^(-3)s$. What is the oscillation frequency in this case? What is the cyclic frequency of vibration of this mass?
Solution. The oscillation frequency is the reciprocal of the oscillation period, therefore, to solve the problem it is enough to use the formula:
\[\nu =\frac(1)(T)\left(1.1\right).\]
Let's calculate the required frequency:
\[\nu =\frac(1)(5\cdot (10)^(-3))=200\ \left(Hz\right).\]
The cyclic frequency is related to the frequency $\nu $ as:
\[(\omega )_0=2\pi \nu \ \left(1.2\right).\]
Let's calculate the cyclic frequency:
\[(\omega )_0=2\pi \cdot 200\approx 1256\ \left(\frac(rad)(s)\right).\]
Answer.$1)\ \nu =200$ Hz. 2) $(\omega )_0=1256\ \frac(rad)(s)$
Example 2
Exercise. The mass of the load hanging on an elastic spring (Fig. 2) is increased by $\Delta m$, while the frequency decreases by $n$ times. What is the mass of the first load?
\[\nu =\frac(1)(2\pi )\sqrt(\frac(k)(m))\ \left(2.1\right).\]
For the first load the frequency will be equal to:
\[(\nu )_1=\frac(1)(2\pi )\sqrt(\frac(k)(m))\ \left(2.2\right).\]
For the second load:
\[(\nu )_2=\frac(1)(2\pi )\sqrt(\frac(k)(m+\Delta m))\ \left(2.2\right).\]
According to the conditions of the problem $(\nu )_2=\frac((\nu )_1)(n)$, we find the relation $\frac((\nu )_1)((\nu )_2):\frac((\nu )_1)((\nu )_2)=\sqrt(\frac(k)(m)\cdot \frac(m+\Delta m)(k))=\sqrt(1+\frac(\Delta m)( m))=n\ \left(2.3\right).$
Let us obtain from equation (2.3) the required mass of the load. To do this, let’s square both sides of expression (2.3) and express $m$:
Answer.$m=\frac(\Delta m)(n^2-1)$
A physical system (body) in which oscillations arise and exist when deviating from the equilibrium position is called oscillatory system.
Let's consider the simplest mechanical oscillatory systems: spring and mathematical pendulums.
Spring pendulum
- Spring pendulum is an oscillatory system consisting of a material point of mass m and a spring.
Distinguish horizontal spring pendulum (Fig. 1, a) and vertical(Fig. 1, b).
The period of oscillation of a spring pendulum can be found using the formula
\(T=2\pi \cdot \sqrt(\frac(m)(k)),\)
Where k- coefficient of rigidity of the pendulum spring. As follows from the formula obtained, the period of oscillation of a spring pendulum does not depend on the amplitude of oscillations (within the limits of the feasibility of Hooke's law).
- The property of independence of the period of oscillation of a pendulum from the amplitude, discovered by Galileo, is called isochronicity(from the Greek words ίσος - equal and χρόνος - time).
Math pendulum
Consider a simple pendulum - a ball suspended on a long strong thread. Such a pendulum is called physical.
If the dimensions of the ball are much smaller than the length of the thread, then these dimensions can be neglected and the ball can be considered as a material point. The stretch of the thread can also be neglected, since it is very small. If the mass of the thread is many times less than the mass of the ball, then the mass of the thread can also be neglected. In this case, we get a model of a pendulum, which is called a mathematical pendulum.
- Mathematical pendulum is called a material point of mass m, suspended on a weightless inextensible thread of length l in the field of gravity (or other forces) (Fig. 2).
Galileo Galilei experimentally established that the period of oscillation of a mathematical pendulum in a gravity field does not depend on its mass and amplitude of oscillations (the angle of initial deflection). He also established that the period of oscillation is directly proportional to \(\sqrt(l)\).
The period of small oscillations of a mathematical pendulum in the Earth's gravity field is determined by Huygens' formula:
\(T=2\pi \cdot \sqrt(\frac(l)(g)).\)
At angles of deflection of the mathematical pendulum α< 20° погрешность расчета периода по формуле Гюйгенса не превышает 1%.
In the general case, when the pendulum is in uniform fields of several forces, then to determine the period of oscillation one should enter “ efficient acceleration» g*, characterizing the resulting action of these fields and the period of oscillation of the pendulum will be determined by the formula
\(T=2\pi \cdot \sqrt(\frac(l)(g*)).\)
*Derivation of formulas
*Spring pendulum
For cargo m horizontal spring pendulum is subject to the force of gravity ( m⋅g), ground reaction force ( N) and the elastic force of the spring ( Fynp) (Fig. 3, the first two forces in Fig. A not specified). Let us write down Newton's second law for the case shown in Fig. 3, b
\(m\cdot \vec(a) = \vec(F)_(ynp) + m\cdot \vec(g)+\vec(N),\)
0X\ or \(m\cdot a_(x) +k\cdot x=0.\)
Rice. 3.
Let us write this equation in a form similar to the equation of motion of a harmonic oscillator
\(a_(x) + \frac(k)(m) \cdot x = 0.\)
Comparing the resulting expression with the equation of harmonic vibrations
\(a_(x) (t) + \omega^(2) \cdot x(t) = 0,\)
find the cyclic frequency of oscillations of the spring pendulum
\(\omega = \sqrt(\frac(k)(m)).\)
Then the period of oscillation of the spring pendulum will be equal to:
\(T=\frac(2\pi )(\omega ) = 2\pi \cdot \sqrt(\frac(m)(k)).\)
*Mathematical pendulum
For cargo m mathematical pendulum acted by gravity ( m⋅g) and the elastic force of the thread ( Fynp) (tension force) (Fig. 4). Axis 0 X Let's direct it along the tangent to the trajectory of upward movement. Let us write down Newton's second law for the case shown in Fig. 4, b
\(m\cdot \vec(a) = \vec(F)_(ynp) + m\cdot \vec(g),\)
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, i.e. 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 in the form
This formula expresses natural frequency of small oscillations of a mathematical pendulum .
Hence,
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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:
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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 the 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
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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 an 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. Attached to the upper end of the pendulum 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 talking or singing, etc.
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| Figure 2.5.4. Clock mechanism with a pendulum. |
