To describe the oscillatory processes and distinguish some oscillations from others, use 6 characteristics. They are called so (Fig. 1):
- amplitude,
- period,
- frequency,
- cyclic frequency
- phase,
- The initial phase.
Fig. 1. The main characteristics of oscillations are amplitude, period and initial phase
Such values as amplitude and period can be determined by chart of oscillations.
The initial phase is also determined by the schedule, using the time interval \ (\ large \ delta t \), to which relative to zero is shifted by the beginning of the nearest period.
The frequency and cyclic frequency are calculated from the period found according to the formulas. They are below the text of this article.
And the phase is determined by the formula into which the time of interest is interested in the time of T oscillations. Read further.
What is amplitude
The amplitude is the greatest deviation of the value from equilibrium, that is, the maximum value of the oscillating value.
Measure in the same units in which the oscillating value is measured. For example, when we consider mechanical oscillations in which the coordinate changes, the amplitude is measured in meters.
In the case of electrical oscillations in which the charge changes, it is measured in the coulons. If current fluctuates in amperes, and if there is a voltage, then in volts.
Often designate it, attributing to the letter denoting an amplitude index "0" from below.
For example, let the magnitude \ (\ Large X \). Then the \ (\ large x_ {0} \) symbol denote the amplitude of the oscillations of this value.
Sometimes, to designate amplitudes, a large Latin letter A is used, as this is the first letter of the English word "amplitude".
Using the graph, the amplitude can be determined so (Fig. 2):
Fig. 2. The amplitude is the maximum deviation from the horizontal axis or up, or down. The horizontal axis passes through the level of zero on the axis, which marks amplitudes
What is a period
When the oscillations are repeated exactly, the changing value takes the same values through the same pieces of time. Such a piece of time is called a period.
Indicate it usually a large Latin letter "T" and is measured in seconds.
\ (\ LARGE T \ Left (C \ Right) \) - period of oscillations.
One second is a fairly large time interval. Therefore, although the period is measured in seconds, but for most oscillations it will be measured by shares of a second.
To determine the vibration schedule to determine the period (Fig. 3), you need to find two identical values of the oscillating value. After, spending from these values to the dotted time axis. The distance between dosses is a period of oscillations.
Fig. 3. Period of oscillations - this is a horizontal distance between two similar points on the chart
The period is the time of one complete oscillation.
On the chart, the period is more convenient to find one of these ways (Fig. 4):
Fig. 4. It is convenient to determine the period as the distance between two adjacent vertices, or between two depressions
What is frequency
Denote it with the help of the Greek letter "NU" \ (\ LARGE \ NU \).
The frequency answers the question: "How many full oscillations are performed in one second?" Or: "How many periods fits at the time interval equal to one second?".
Therefore, the dimensionality of the frequency is the vibration units per second:
\ (\ Large \ Nu \ Left (\ FRAC {1} {C} \ Right) \).
Sometimes in the textbooks there is such an entry \ (\ large \ displaystyle \ nu \ left (C ^ {- 1} \ Right) \), because according to the degree properties \ (\ large \ displaystyle \ FRAC {1} {C} = C ^ {- 1} \).
Since 1933, the frequency is indicated in Hertz in honor of Herrich Rudolph Hertz. He committed significant discoveries in physics, studied oscillations and proved that electromagnetic waves exist.
One oscillation per second corresponds to the frequency of 1 hertz.
\ [\ Large \ DisplayStyle \ Boxed {\ FRAC {1 \ TEXT {{}}} {1 \ Text {second}} = 1 \ Text {Hz}} \]
To determine the frequency using the graph, it is necessary to determine the period in the time axis. And then calculate the frequency of such a formula:
\ [\ LARGE \ BOXED {\ NU = \ FRAC {1} {t}} \]
There is another way to determine the frequency using the graph of the oscillating value. You need to measure the time interval in the chart equal to one second, and to count the number of periods of oscillations that were relevant to this interval (Fig. 5).
Fig. 5. On the chart the frequency is the number of periods that have relevant in one second
What is cyclic frequency
The oscillatory movement and the movement around the circle have a lot of common - these are repeated movements. One full turn corresponds to the angle \ (\ large 2 \ pi \) radian. Therefore, in addition to the time interval of 1 second, physicists use the time interval equal to \ (\ LARGE 2 \ PI \) seconds.
The number of complete oscillations for such a time interval is called cyclic frequency and is indicated by the Greek letter "Omega":
\ (\ Large \ DisplayStyle \ Omega \ Left (\ FRAC {\ Text {RF}} {C} \ Right) \)
Note: The value \ (\ large \ omega \) is also called a circular frequency, and also - an angular speed (link).
Cyclic frequency answers the question: "How many full oscillations are performed for \ (\ large 2 \ pi \) seconds?" Or: "How many periods fit at the time interval equal to \ (\ large 2 \ pi \) seconds?".
The usual \ (\ large \ nu \) and cyclic \ (\ large \ omega \) The frequency of oscillations are related to the formula:
\ [\ Large \ Boxed {\ omega = 2 \ pi \ cdot \ nu} \]
On the left in the formula, the amount of oscillations is measured in radians for a second, and on the right - in the hertz.
To determine the value of \ (\ large \ omega \) using the oscillation schedule, you must first find the period T.
Then, use the formula \ (\ large \ displaystyle \ nu = \ frac {1} {t} \) and calculate the frequency \ (\ large \ nu \).
And only after that, with the help of formula \ (\ large \ omega = 2 \ pi \ cdot \ Nu \), calculate the cyclic \ (\ large \ omega \) frequency.
For a rough oral assessment, we can assume that the cyclic frequency exceeds the usual frequency of about 6 times numerically.
Determine the value \ (\ large \ omega \) according to the vibration schedule is still in one way. On the time axis, the interval equal to \ (\ large 2 \ pi \), and then, count the number of periods of oscillations in this interval (Fig. 6).
Fig. 6. On the chart of cyclic (circular) frequency - this is the number of periods that were relevant in 2 pi seconds
What is the initial phase and how to determine it according to the vibration schedule
I will reject the swing at some angle of equilibrium and will hold them in this position. When we let go, the swings will begin to swing. And the start of the oscillations will occur from the corner to which we rejected them.
Such, the initial angle of deviation is called the initial phase of oscillations. Denote this angle (Fig. 7) of some Greek letter, for example, \ (\ Large \ Varphi_ {0} \).
\ (\ large \ varphi_ {0} \ left (\ text {rad} \ right) \) - the initial phase, is measured in radians (or degrees).
The initial phase of oscillations is the angle on which we rejected the swing before letting them go. From this angle will begin the oscillating process.
Fig. 7. The angle of deviation of the swing before the start of oscillations
Consider now how the value \ (\ large \ varphi_ {0} \) affects the vibration schedule (Fig. 8). For convenience, we assume that we consider the oscillations that occur by the law of sinus.
The curve marked with black in the figure begins the period of oscillations from the point T = 0. This curve is a "clean", not shifted by sine. For it, the magnitude of the initial phase \ (\ large \ varphi_ {0} \) is taken equal to zero.
Fig. 8. The vertical position of the start point at time t = 0 and the shift of the horizontal graph is determined by the initial phase
The second curve in the picture is marked in red. The beginning of its period is shifted to the right relative to the point T = 0. Therefore, for a red curve, which began a new period of oscillations after time \ (\ Large \ Delta T \), the initial angle \ (\ Large \ Varphi_ {0} \) will differ from zero values.
We define the angle \ (\ large \ varphi_ {0} \) using the oscillation schedule.
We draw attention (Fig. 8) to the fact that the time lying on the horizontal axis is measured in seconds, and the value \ (\ large \ varphi_ {0} \) - in radians. So, you need to link a formula of a piece of time \ (\ large \ delta t \) and the initial angle corresponding to it \ (\ large \ varphi_ {0} \).
How to calculate the initial angle on the interval of offset
The algorithm for finding an initial angle consists of several uncomplicated steps.
- First, we define the time interval marked with blue arrows in the picture. On the axes of most charts there are numbers for which it can be done. As can be seen from fig. 8, this interval \ (\ Large \ Delta T \) is 1 sec.
- Then we define the period. To do this, we note one complete oscillation on the red curve. The oscillation began at the point T = 1, and it ended at point T = 5. Taking the difference between these two points of time, we obtain the value of the period.
\ [\ Large T = 5 - 1 = 4 \ Left (\ Text {s} \ Right) \]
From the graph, it follows that the period T = 4 seconds.
- Calculate now, what fraction of the period is the time interval \ (\ LARGE \ Delta T \). To do this, we will make such a fraction \ (\ large \ displaystyle \ FRAC {\ Delta T} {t} \):
\ [\ LARGE \ FRAC {\ Delta T} {t} = \ FRAC {1} {4} \]
The resulting fraction value means that the red curve is shifted relative to the point T = 0 and the black curve by a quarter of the period.
- We know that one complete oscillation is one full turn (cycle), sinus (or cosine) performs, passing each time an angle \ (\ large 2 \ pi \). We now find how the found share of the period with an angle \ (\ large 2 \ pi \) is associated with the full cycle.
To do this, use the formula:
\ [\ Large \ Boxed {\ FRAC {\ Delta T} {t} \ Cdot 2 \ pi = \ varphi_ {0}} \]
\ (\ Large \ DisplayStyle \ FRAC {1} {4} \ cdot 2 \ pi = \ frac {\ pi} {2} = \ varphi_ {0} \)
So, the interval \ (\ Large \ Delta T \) corresponds to the angle \ (\ large \ displaystyle \ frac {\ pi} {2} \) is the initial phase for the red curve in the figure.
- In conclusion, pay attention to the following. The beginning of the nearest to point T = 0 period of the red curve is shifted to the right. That is, the curve delays relative to the "clean" sine.
To designate delay, we will use the minus sign for the initial angle:
\ [\ Large \ Varphi_ {0} = - \ FRAC {\ PI} {2} \]
Note: If on the oscillation curve, the beginning of the nearest period is the left of the point T = 0, then in this case, the angle \ (\ large \ displaystyle \ FRAC {\ pi} {2} \) has a plus sign.
For not shifted to the left, either right, sinus or cosine, the initial phase of zero \ (\ large \ varphi_ {0} = 0 \).
For sinus or cosine, shifted to the left in graphics and ahead of the usual function, the initial phase is taken with the "+" sign.
And if the function is shifted to the right and delays relative to the usual function, the value \ (\ large \ varphi_ {0} \) is written with the "-" sign.
Notes:
- Physicists begin countdown from point 0. Therefore, time in tasks is not negative.
- On the chart of oscillations, the initial phase \ (\ varphi_ {0} \) affects the vertical shift of the point from which the oscillating process starts. So, it is possible to say that oscillations have a starting point.
Thanks to such assumptions, the vibration schedule in solving most tasks can be depicted, starting from the neighborhood of zero and mainly in the right half-plane.
What is the oscillation phase
Consider once again ordinary children's swings (Fig. 9) and the angle of their deviation from the equilibrium position. Over time, this angle varies, that is, it depends on time.
Fig. 9. The angle of deviation from equilibrium - phase, changes in the process of oscillations
In the process of oscillations, an angle of deviation from equilibrium changes. This changing angle is called the oscillation phase and denote \ (\ varphi \).
Differences between phase and initial phase
There are two angle deviations from equilibrium - initial, it is set before the start of oscillations and, the angle that changes during the oscillations.
The first angle is called the initial \ (\ varphi_ {0} \) phase (Fig. 10a), it is considered to be unchanged. And the second angle is simply \ (\ varphi \) a phase (Fig. 10b) is the value of the variable.
Fig. 10. Before starting the oscillations, we specify the initial phase - the initial angle of deviation from equilibrium. And the angle that changes during the oscillations is called a phase
As on the chart of oscillations to mark the phase
On the chart of oscillations of the phase \ (\ large \ varphi \) looks like a point on the curve. Over time, this point is shifted (running) on schedule from left to right (Fig. 11). That is, at different points in time it will be in different parts of the curve.
The figure marked two large red dots, they correspond to the oscillation phases at times T1 and T2.
Fig. 11. On the chart of the oscillations of the phase is a point that sliding on the curve. At various points in time, it is in different positions on the chart.
And the initial phase on the chart of oscillations looks like a place where the point lying on the oscillation curve is at time t = 0. The figure additionally contains one small red dot, it corresponds to the initial oscillation phase.
How to determine the phase using the formula
Let us know the magnitude \ (\ large \ omega \) - the cyclic frequency and \ (\ large \ varphi_ {0} \) - the initial phase. During the oscillations, these values do not change, that is, are constants.
Time oscillations T will be a variable value.
The phase \ (\ large \ varphi \), corresponding to any time of interest to us, can be determined from such an equation:
\ [\ Large \ Boxed {\ Varphi = \ Omega \ CDOT T + \ VARPHI_ {0}} \]
The left and right parts of this equation have the dimension of the angle (i.e. they are measured in radians, or degrees). And substituting instead of a symbol T into this equation of the time you are interested in, you can get the corresponding phase values.
What is the phase difference
Usually the concept of phase difference is used when they compare two oscillatory process among themselves.
Consider two oscillatory process (Fig. 12). Each has its initial phase.
Denote them:
\ (\ large \ varphi_ {01} \) - for the first process and,
\ (\ Large \ Varphi_ {02} \) - for the second process.
Fig. 12. For two oscillations, you can enter the concept of phase difference
We define the phase difference between the first and second oscillatory processes:
\ [\ Large \ Boxed {\ Delta \ Varphi = \ Varphi_ {01} - \ varphi_ {02}} \]
The value \ (\ Large \ Delta \ Varphi \) shows how many phases of two oscillations are distinguished, it is called the phase difference.
How are the characteristics of oscillations - formulas
Movement around the circle and oscillatory movement have a certain similarity, since these types of movement can be periodic.
Therefore, the basic formulas applicable to the circle movement will also fit the same to describe the oscillatory movement.
- The relationship between the period, the amount of oscillations and the total time of the oscillatory process:
\ [\ Large \ Boxed {t \ cdot n = t} \]
\ (\ LARGE T \ LEFT (C \ Right) \) - the time of one complete oscillation (period of oscillations);
\ (\ LARGE N \ Left (\ Text {pieces} \ Right) \) - the number of complete oscillations;
\ (\ Large T \ Left (C \ Right) \) - total time for several oscillations;
- The period and frequency of oscillations are associated as:
\ [\ Large \ Boxed {T = \ FRAC {1} {\ Nu}} \]
\ (\ Large \ Nu \ Left (\ Text {Hz} \ Right) \) - frequency of oscillations.
- The amount and frequency of oscillations are related to the formula:
\ [\ Large \ Boxed {n = \ Nu \ Cdot T} \]
- Communication between the frequency and cyclic frequency of oscillations:
\ [\ Large \ Boxed {\ Nu \ Cdot 2 \ pi = \ omega} \]
\ (\ Large \ DisplayStyle \ Omega \ Left (\ FRAC {\ Text {Right}} {C} \ Right) \) - cyclic (circular) oscillation frequency.
- Phase and cyclic oscillation frequency are associated as follows:
\ [\ Large \ Boxed {\ Varphi = \ Omega \ CDOT T + \ VARPHI_ {0}} \]
\ (\ large \ varphi_ {0} \ left (\ text {rad} \ right) \) - the initial phase;
\ (\ large \ varphi \ left (\ text {rad} \ right) \) - phase (angle) at the selected time t;
- Between the phase and the amount of oscillations, the link is described as:
\ [\ Large \ Boxed {\ varphi = n \ cdot 2 \ pi} \]
- The time interval \ (\ Large \ Delta T \) (shift) and the initial phase of oscillations are related:
\ [\ Large \ Boxed {\ FRAC {\ Delta T} {t} \ Cdot 2 \ pi = \ varphi_ {0}} \]
\ (\ Large \ Delta T \ Left (C \ Right) \) - the time interval on which relative to the point T = 0 shifted the beginning of the nearest period.
Consider the values by which you can characterize oscillations.
Compare oscillations of two swings in the picture - empty swings and swings with a boy. Swing with a boy fluctuate with a big sweep, that is, their extreme positions are further from the equilibrium position than that of empty swing.
The largest (module) deviation of the oscillating body on the position of the equilibrium is called the amplitude of the oscillations.
Pay attention!
The amplitude of oscillations, as a rule, is denoted by the letter \ (A \) and in Xi is measured in meters (m).
Example:
Pay attention!
The amplitude can also be measured in units of a flat angle, for example in degrees, since the circumferential arc corresponds to a certain central angle, that is, angle with a vertex in the center of the circle.
The oscillating body makes one complete oscillation if a path equal to four amplitudes passes from the beginning of the oscillations.
The period of time during which the body makes one complete oscillation, is called a period of oscillations.
Pay attention!
The period of oscillations is denoted by the letter \ (T \) and in Si is measured in seconds (C).
Example:
I will hit the table with two rules - metal and wooden. The line after that will begin to fluctuate, but in the same time the metal line (a) will make more oscillations than the wooden (B).
The number of oscillations per unit of time is called the frequency of oscillations.
Pay attention!
Denotes the frequency of the Greek letter $\nu $("NU"). Per unit of frequency accepted one oscillation per second. This unit in honor of the German scientist Henry Hertz is named Hertz (Hz).
Oscillation period \ (T \) and oscillation frequency $\nu $related to the following dependence:
$\mathrm{T.}=\frac{1}{\nu}$.
Free oscillations in the absence of friction and resistance of air are called their own oscillations, and their frequency is their own frequency of the oscillating system.
Any oscillatory system has a specific one's own frequency depending on the parameters of this system. For example, the proprietary frequency of the spring pendulum depends on the mass of the cargo and the rigidity of the spring.
Consider the oscillations of two identical empty swings in the figure above. In the same time, the red swings from the equilibrium position begin forward moving, and the green swings from the equilibrium position move back. Swing fluctuate with the same frequency and with the same amplitudes. However, these oscillations differ from each other: at any time the speed of the swings are directed in opposite sides. In this case, they say that swing oscillations occur in opposite phases.
Red empty swings and swings with a boy also fluctuate with the same frequencies. The speed of these swings at any time is directed equally. In this case, they say that the swing fluctuate in the same phases.
The physical value, called phase, is used not only when comparing the oscillations of two or more bodies, but also to describe the oscillations of one body.
Thus, the oscillatory movement is characterized by an amplitude, frequency (or period) and phase.
Sources:
Physics. 9 CL.: Tutorial / Pryrickin A. V., Godnik E. M. - M.: Drop, 2014. - 319 S.www.ru.Depositphotos.com, Site "Photobank with a premium collection of photos, vectors and video"
www.mognovse.ru, the site "You can all"
The work of most mechanisms is based on the simplest laws of physics and mathematics. A rather large distribution received the concept of a spring pendulum. Such a mechanism was obtained very widespread, since the spring provides the required functionality, it may be an element of automatic devices. Consider a similar device, the principle of operation and many other points in more detail.
Spring pendulum definitions
As previously noted, the spring pendulum was obtained very widespread. Among the features, you can note the following:
- The device is represented by a combination of cargo and springs, the mass of which may not be taken into account. As a cargo, the most different object can be. At the same time, it may be affected by external force. A common example can be called the creation of a safety valve that is installed in the pipeline system. The cargo mounting to the spring is carried out in the most different way. It uses an exceptionally classic screw version that has become the most widespread. The main properties are largely dependent on the type of material used in the manufacture, the diameter of the turn, the correctness of the centering and many other points. The extreme turns are often manufactured in such a way as to perceive a large load during operation.
- Prior to the start of deformation, there is no complete mechanical energy. At the same time, the power of elasticity does not affect the body. Each spring has an initial position that it retains for a long period. However, due to certain rigidity, body fixation occurs in the initial position. It matters how the effort is applied. An example is that it should be directed along the springs axis, since otherwise there is a possibility of deformation and many other problems. Each spring has its own definite compression and stretching. At the same time, the maximum compression is represented by the absence of a gap between individual turns, when tensioning there is a moment when the irrevocative deformation of the product occurs. With too much elongation, the wire changes the basic properties, after which the product does not return to its original position.
- In the case under consideration, the oscillations are made due to the action of the force of elasticity. It is characterized by a rather large number of features that should be taken into account. The impact of elasticity is achieved due to a certain arrangement of turns and the type of material used in the manufacture. At the same time, the power of elasticity can act in both directions. Most often compressed, but it can also be stretched - it all depends on the characteristics of a particular case.
- The speed of the movement of the body can vary in a sufficiently large range, it all depends on what is the impact. For example, the spring pendulum can move the suspended cargo in the horizontal and vertical plane. The action of aimed force depends largely on the vertical or horizontal installation.
In general, we can say that the spring pendulum definition is rather generalized. In this case, the speed of movement of an object depends on various parameters, for example, the values of the applied force and other points. The direct settlement of the calculations is the creation of a scheme:
- Specifies the support to which the spring is attached. Often for its display drawn a line with reverse hatching.
- Schematically displays a spring. It is presented by a wavy line. During a schematic mapping, the length and diametrical indicator does not matter.
- Also depicted body. It should not match the sizes, however, it matters the place of direct attachment.
The scheme is required for a schematic display of all forces that affect the device. Only in this case can be taken into account everything that affects the speed of movement, inertia and many other points.
Spring pendulums are applied not only when calculating the silt solutions of various tasks, but also in practice. However, not all properties of such a mechanism are applicable.
An example can be called a case when oscillatory movements are not required:
- Creating shut-off elements.
- Spring mechanisms associated with transportation of various materials and objects.
The spent 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:
- Diameter of turns. It may be the most different. The diameter indicator largely depends on how much the material is required for production. The diameter of turns also defines how much effort should be applied to complete compression or partial stretching. However, the increase in dimensions can create significant difficulties with the installation of the product.
- The diameter of the wire. Another important parameter can be called the diametrical size of the wire. It can vary in a wide range, the strength and degree of elasticity depends.
- Length of the product. This indicator determines what effort is required for complete compression, as well as the product may have a product.
- The type of material used also determines the basic properties. Most often, the spring is manufactured when applying a special alloy, which has the corresponding properties.
With mathematical calculations, many points are not taken into account. Elastic force and many other indicators are detected by calculation.
Types of spring pendulum
Several different types of spring pendulum are distinguished. It should be borne in mind that the classification can be carried out by the type of springs installed. Among the features, we note:
- Vertical oscillations received quite a lot of distribution, since in this case, friction force and other impact are not on the cargo. With the vertical location of the cargo, the degree of gravity force is significantly increasing. This version of execution is distributed when conducting a wide variety of calculations. Due to the gravity, there is a possibility that the body at the starting point will perform a large amount of inertial movements. This also contributes to the elasticity and inertia of the body movement at the end of the course.
- Also used horizontal spring pendulum. In this case, the cargo is located on the supporting surface and friction also occurs at the time of movement. With a horizontal arrangement, the gravity strength works somewhat differently. The horizontal body location was widespread in various tasks.
The movement of the spring pendulum can be calculated when using a sufficiently large number of different formulas, which should take into account the impact of all forces. In most cases, a classic spring is installed. Among the features, we note the following:
- The classic twisted compression spring today was widely widespread. In this case, there is a space between the turns that is called a step. The compression spring can and stretch, but it is often 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 the uniform distribution of the effort is ensured.
- An embodiment can be installed for stretching. It is designed to be installed in the case when the applied force causes an increase in length. For fasteners, hooks are accommodated.
Completed both options. It is important to pay attention to the fact that the force is applied parallel to the axis. Otherwise, there is a possibility of turning the turns that it becomes causes serious problems, for example, deformation.
The strength of elasticity in the spring pendulum
It is necessary to take into account the moment that before deformation of the spring it is in the equilibrium position. The applied force can lead to its stretching and compressing. The strength of elasticity in the spring pendulum is calculated in accordance with how the law of energy conservation is affected. According to adopted standards, the elasticity arising is proportional to the bias. In this case, the kinetic energy is calculated by the formula: F = -KX. In this case, the coefficient of spring is applied.
A rather large number of features of the effect of elasticity in the spring pendulum are distinguished. Among the features, we note:
- The maximum force of elasticity occurs at the time when the body is at the maximum distance from the equilibrium position. At the same time, in this position, the maximum value of the acceleration of the body is noted. It should not be forgotten that it can be stretched and compression of the spring, both options are somewhat different. When compressed, the minimum length of the product is limited. As a rule, it has a length equal to the diameter of the turn multiplied by the amount. Too much effort can cause turns offset, as well as wire deformations. When tensile, there is a moment of elongation, after which the deformation occurs. Strong elongation leads to the fact that the emergence of elasticity is not enough to return the product to the original state.
- When the body is brought together to the place of equilibrium, there is a significant decrease in the length of the spring. Due to this, there is a constant decrease in the acceleration rate. All this is due to the impact of the effort of elasticity, which is associated with the type of material used in the manufacture of the spring and its features. Length decreases due to the fact that the distance between the turns is reduced. A feature can be called a uniform distribution of turns, only only in case of defects there is a possibility of violation of such a rule.
- At the time of the point of equilibrium, the force of elasticity is reduced to zero. However, the speed is not reduced, as the body moves on inertia. The equilibrium point is characterized by the fact that the length of the product in it is preserved for a long period, subject to the absence of an external deforming force. The equilibrium point is determined in the case of constructing the scheme.
- After reaching the point of equilibrium, the elasticity arising begins to reduce the speed of the body movement. It acts in the opposite direction. In this case, an effort occurs, which is directed in the opposite direction.
- Having reached the extreme point of the body begins to move in the opposite direction. Depending on the rigidity of the installed spring, this action will be repeated repeatedly. The length of this cycle depends on the most different points. An example can be called a body weight, as well as the maximum applied force for the occurrence of deformation. In some cases, the oscillatory movements are practically invisible, but they still arise.
The above information indicates that the oscillatory movements are made due to the effects of elasticity. The deformation occurs due to the applied effort, which can vary in a sufficiently large range, it all depends on the specific case.
Spring pendulum oscillation equations
The fluctuations of the spring pendulum are committed by harmonious law. The formula for which the calculation is carried out is as follows: f (t) = Ma (T) = - MW2x (T).
The above formula indicates (W) the radial frequency of harmonic oscillation. It is characteristic of strength, which spreads within the limits of the applicability of the bike law. The motion equation can differ significantly, it all depends on the specific case.
If we consider the oscillatory movement, then the following points should be given:
- The oscillatory movements are observed only at the end of the body movement. Initially, it is straightforward to the complete liberation of effort. At the same time, the force of elasticity is maintained throughout the entire time until the body is in the maximum remote position from zero coordinates.
- After stretching the body returns to its original position. The emerging inertia becomes the reason for which the exposure to the spring can be provided. Inertia largely depends on body weight, advanced speed and many other points.
As a result, a oscillation occurs, which can last for a long period. The above formula allows you to calculate with all the moments.
Formulas period and frequency of fluctuations of spring pendulum
When designing and calculating the main indicators, quite a lot of attention is paid to the frequency and period of oscillation. Cosine is a periodic function in which the value is applied unchanged after a certain period of time. This indicator calls the period of fluctuations in the spring pendulum. To refer to this indicator, the letter T is used, the concept characterizers the reverse period of oscillation (V) is also often used. In most cases, in the calculations, the formula T = 1 / V is used.
The oscillation period is calculated in a somewhat complicated formula. It is as follows: t = 2p√m / k. To determine the oscillation frequency, the formula is used: V = 1 / 2P√K / M.
The cyclic frequency of the fluctuations in the spring pendulum depends on the following points:
- The weight of the cargo that is attached to the spring. This indicator is considered the most important, as it affects the most different parameters. Mass depends the power of inertia, speed and many other indicators. In addition, the weight of the cargo is the value, with the measurement of which there is no problems due to the presence of special measuring equipment.
- The coefficient of elasticity. For each spring, this figure is significantly different. The elastic coefficient 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 the most different way, often when applying special equipment.
Do not forget that with a strong stretching of the spring, the law of the thief stops acting. At the same time, the period of spring oscillation begins to depend on the amplitude.
To measure the period, the World Unit of Time is used, in most cases seconds. In most cases, the amplitude of oscillations is calculated when solving a variety of tasks. To simplify the process, a simplified scheme is based on, which displays the main forces.
Amplitude formulas and the initial phase of the spring pendulum
Deciding with the peculiarities of passable processes and knowing the oscillation equation of the spring pendulum, as well as the initial values of the amplitude and the initial phase of the spring pendulum. To determine the initial phase, the value f is applied, the amplitude is indicated by the symbol A.
To determine the amplitude, the formula can be used: a = √x ^{2}+ V. ^{2}/ W. ^{2}. The initial phase is calculated by the formula: TGF = -V / XW.
Applying these formulas can be determined by the basic parameters that are used in the calculations.
Energy of spring pendulum oscillations
Considering the oscillation of the cargo on the spring, it is necessary to take into account the moment that when moving the pendulum can be described by two points, that is, it is rectilinear. This moment determines the fulfillment of the conditions relating to the force under consideration. It can be said that the total energy is potential.
Conduct the calculation of the energy of the oscillations of the spring pendulum can be taken into account by all features. The main points will call the following:
- Oscillations can be held in a horizontal and vertical plane.
- The zero of potential energy is chosen as an equilibrium position. It is in this place that the origin of coordinates is established. As a rule, in this position, the spring retains its shape under the condition of the absence of deforming force.
- In the case under consideration, the calculated energy of the spring pendulum does not take into account the force of friction. With a vertical location of the cargo, the friction force is insignificant, with a horizontal body is on the surface and friction may occur when moving.
- To calculate the oscillation energy, the following formula is used: E = -DF / DX.
The above information indicates that the law of energy conservation is as follows: MX ^{2}/ 2 + MW ^{2}X. ^{2}/ 2 = const. The formula applied is as follows:
- The maximum kinetic energy of the installed pendulum is directly proportional to the maximum potential value.
- At the time of the oscillator, the average value of both strength is equal.
Conduct the determination of the energy of the spring pendulum fluctuations in solving a variety of tasks.
Free fluctuations in spring pendulum
Considering what the free fluctuations of the spring pendulum are caused by the action of the internal forces. They begin to form almost immediately after the body was transmitted. Features of harmonic oscillations are included in the following points:
- Other types of affecting forces may also arise, which satisfies all the norms of the law, are called quasi-elastic.
- The main reasons for the action of the law may be internal forces that are formed directly at the time of changing the position of the body in space. At the same time, the cargo has a certain mass, the force is created by fixing one end for a fixed object with sufficient strength, the second for the goods itself. Subject to the absence of friction, the body can perform oscillatory movements. In this case, the fixed load is called linear.
You should not forget that there is simply a huge number of different types of systems in which a oscillatory movement is carried out. They also arise to elastic deformation, which becomes the cause of application for performing any work.
The main formulas in physics - oscillations and waves
When studying this section should be borne in mind that oscillations Various physical nature is described with uniform mathematical positions. Here it is necessary to clearly understand the concepts such as harmonic oscillation, phase, phase difference, amplitude, frequency, period of oscillations.
It should be borne in mind that in any real oscillatory system there are resistances of the medium, i.e. The oscillations will be attenuating. To characterize the attenuation of oscillations, the attenuation coefficient and the logarithmic decrement of the atuchi are injected.
If oscillations are performed under the action of an external periodically changing force, then such oscillations are called forced. They will be unsuccessful. The amplitude of the forced oscillations depends on the frequency of the forcing force. When the frequency of forced oscillations approaches the frequency of its own oscillations of the amplitude of the forced oscillations increases sharply. This phenomenon is called resonance.
Moving to the study of electromagnetic waves need to clearly represent that Electromagnetic wave - This is an electromagnetic field spreading in space. The simplest system emitting electromagnetic waves is an electric dipole. If the dipole performs harmonic oscillations, then it emits a monochromatic wave.
See also the basic formulas of quantum physics
Table of formulas: oscillations and waves
Physical laws, formulas, variables | Formulas of oscillations and waves | ||||
Harmonic oscillation equation: where x - offset (deviation) of the oscillating value from the equilibrium position; A - amplitude; ω - circular (cyclic) frequency; t - time; α - initial phase; (ωt + α) - phase. | |||||
Communication between the period and circular frequency: | |||||
Frequency: | |||||
Circular frequency connection with frequency: | |||||
Periods of own oscillations 1) Spring pendulum: where k is the rigidity of the spring; 2) Mathematical pendulum: where L is the length of the pendulum, g - acceleration of free fall; 3) oscillatory circuit: where L is the inductance of the contour, C - capacitance of the capacitor. | |||||
Frequency of own oscillations: | |||||
Addition of oscillations of the same frequency and direction: 1) the amplitude of the resulting oscillation Where am _{1}and A. _{2}- amplitudes of components of oscillations, α_{1}and α. _{2}- the initial phases of the components of the oscillations; 2) the initial phase of the resulting oscillation |
| ||||
Flowing oscillation equations: E = 2.71 ... - The basis of natural logarithms. | |||||
Sleeping oscillation amplitudes: Where am _{0}- amplitude at the initial moment of time; β - attenuation coefficient; T - Time. | |||||
Attenuation coefficient: Ibitable body where R is the coefficient of resistance of the medium, m - body weight; oscillatory circuit where R is active resistance, L - inductance of the contour. | |||||
Frequency of floating oscillations Ω: | |||||
Period of floating oscillations T: | |||||
Logarithmic decrement attenuation: | |||||
Communication of the logarithmic decrement χ and the attenuation coefficient β: | |||||
The amplitude of forced oscillations where Ω is the frequency of forced oscillations, f_{о}- reduced amplitude forgoing force, With mechanical oscillations: With electromagnetic oscillations: | |||||
Resonant frequency | |||||
Resonant amplitude | |||||
Full oscillation energy: | |||||
Flat Wave Equation: where ξ is the displacement of the points of the medium with the coordinate x at time t; k - wave number: | |||||
Wavelength: where V is the speed of distribution of oscillations in the medium, T - period of oscillations. | |||||
Phase difference relationship Δφ oscillations of two medium points with a distance of Δh between the points of the medium: |
Mechanical oscillations.
Author - Professional tutor, author of textbooks for preparing for the exam
Igor Vyacheslavovich YakovlevThemes of the EGE codifier: harmonic oscillations; amplitude, period, frequency, oscillation phase; Free oscillations, forced oscillations, resonance.
Oscillations - It is repeated in time to change the system status. The concept of oscillations covers a very wide circle of phenomena.
Oscillations of mechanical systems, or Mechanical oscillations - This is a mechanical movement of the body or body system that has a repeatability in time and occurs in the neighborhood of the equilibrium position. Position of equilibrium This state of the system is called in which it can remain as if it is long, without experiencing external influences.
For example, if the pendulum is rejected and release, hesitations will begin. The equilibrium position is the position of the pendulum in the absence of deviation. In this position, the pendulum, if it is not touching it, can be how old. With oscillations, the pendulum passes many times the position of the equilibrium.
Immediately after the rejected pendulum was released, he began to move, the position of the equilibrium passed, reached the opposite of the extreme position, for a moment he stopped in it, moved in the opposite direction, again the position of the equilibrium and returned back. Made one Full oscillation . Further this process will be periodically repeated.
The amplitude of body fluctuations - This is the magnitude of its greatest deviation from the position of equilibrium.
Period of oscillations - This is the time of one complete oscillation. It can be said that for the period the body passes the path of four amplitudes.
Frequency of oscillations - This is the value, reverse period: . The frequency is measured in Hertz (Hz) and shows how many full oscillations are performed in one second.
Harmonic oscillations.
We assume that the position of the oscillating body is determined by a single coordinate
. The position of equilibrium meets the value
. The main task of mechanics in this case is to find a function
giving the coordinate of the body at any time.
For a mathematical description of oscillations, it is natural to use periodic functions. There are many such functions, but two of them are sinus and cosine - are the most important. They have a lot of good properties, and they are closely connected with a wide range of physical phenomena.
Since the functions of sinus and cosine are obtained from each other with a shift of the argument on , It is possible to limit ourselves to one of them. We will use cosine for definition.
Harmonic oscillations - These are oscillations in which the coordinate depends on the time of harmonic law:
(one)
Let's find out the meaning of the magnitudes of this formula.
Positive value It is the largest module with the value of the coordinate (since the maximum value of the cosine module is equal to one), i.e., the greatest deviation from the equilibrium position. therefore - amplitude of oscillations.
Cosine argument called Phase oscillations. Value equal to the value of the phase at , called the initial phase. The initial phase corresponds to the initial coordinate of the body: .
The value is called cyclic frequency . Find her connection with the period of oscillations and frequency . The increment of the phase equal to one complete oscillation radian: From!
(2)
(3)
The cyclic frequency is measured in rad / s (radian per second).
In accordance with expressions (2) и (3) We get two more forms of recording harmonic law (one) :
.
Schedule function (one) , expressing the dependence of the coordinates from time to harmonic oscillations, is shown in Fig. 1.
Fig. 1. Schedule of harmonic oscillations |
Harmonic Vida Law (one) Wears the most common. He responds, for example, situations where two initial acts were performed simultaneously: rejected by the magnitude And they gave him some initial speed. There are two important private events when one of these actions was not committed.
Let the pendulum rejected, but the initial speed was not reported (released without initial speed). It is clear that in this case , so you can put . We get the law of cosine:
.
The graph of harmonic oscillations in this case is shown in Fig. 2.
Fig. 2. Law of Kosinus |
Suppose now that the pendulum was not rejected, but the beacon was informed by the initial speed from the equilibrium position. In this case so you can put . We get the law of sinus:
.
The chart of oscillations is shown in Fig. 3.
Fig. 3. Law of Sinusa |
The equation of harmonic oscillations.
Let's return to the general harmonic law
(one). Differentiating this equality:
. (four)
Now differentiate the beneficial equality (four) :
. (five)
Let's compare expression (one) For coordinates and expression (five) For the projection of acceleration. We see that the projection of acceleration differs from the coordinate only a multiplier :
. (6)
This ratio is called The equation of harmonic oscillations . It can be rewritten and in this form:
. (7)
C mathematical point of view Equation (7) is an Differential equation . Solutions of differential equations serve as functions (and not numbers, as in conventional algebra). So, you can prove that:
- Equation (7) is every function of the form (one) With arbitrary ;
- No other function by solving this equation is not.
In other words, ratios (6) , (7) describe harmonic oscillations with cyclic frequency And only them. Two constants Determined from the initial conditions - according to the initial values of the coordinates and speed.
Spring pendulum.
- This is a load-mounted cargo capable of making fluctuations in a horizontal or vertical direction.
Find a period of small horizontal oscillations of the spring pendulum (Fig. 4). The oscillations will be small if the magnitude of the spring deformation is much less than its size. With small deformations, we can use the leg of the throat. This will lead to the fact that the oscillations will be harmonious.
Friction neglect. The load has a lot , rigid spring is equal .
Coordinate The equilibrium position is responsible, in which the spring is not deformed. Consequently, the magnitude of the springs deformation is equal to the coordinate of the coordinate of the cargo.
Fig. 4. Spring pendulum |
In the horizontal direction on the goods only the force of elasticity is valid From the side of the spring. Newton's second law for cargo in the projection on the axis It has the form:
. (8)
If a (the cargo is shifted to the right, as in the figure), the force of elasticity is directed in the opposite direction, and . On the contrary, if T. . Signs и All the time are opposite, so the law of the knuckle can be written as:
Then the ratio (8) Takes the View:
or
.
We obtained the harmonic oscillation equation of the species (6) , wherein
.
The cyclic frequency of the fluctuations of the spring pendulum is thus equal to:
. (9)
From here and from the ratio We find the period of horizontal fluctuations of the spring pendulum:
. (ten)
If you suspend the load on the spring, the spring pendulum will be obtained, which makes the oscillations in the vertical direction. It can be shown that in this case, for the oscillation period, the formula (ten) .
Mathematical pendulum.
- This is a small body suspended on a weightless non-aggressive thread (Fig.
5). Mathematical pendulum can be fluctuated in the vertical plane in the field of gravity.
Fig. 5. Mathematical pendulum |
Find a period of small oscillations of a mathematical pendulum. The length of the thread is equal . Air resistance neglect.
We write a pendulum Second Newton Law:
,
and we design it on the axis :
.
If the pendulist occupies the position as in the figure (i.e. ), then:
.
If the pendulum is on the other side of the equilibrium position (i.e. ), then:
.
So, at any position of the pendulum, we have:
. (eleven)
When the pendulum rests in the equilibrium position, equality . With low oscillations, when the deviations of the pendulum from the equilibrium position are small (compared with the length of the thread), approximate equality . We use it in the formula (eleven) :
,
or
.
This is the harmonic oscillation equation of the form (6) , wherein
.
Therefore, the cyclic frequency of oscillations of the mathematical pendulum is equal to:
. (12)
Hence the period of oscillations of a mathematical pendulum:
. (thirteen)
Note that in the formula (thirteen) There is no weight of the cargo. Unlike a spring pendulum, the period of oscillations of the mathematical pendulum does not depend on its mass.
Free and forced oscillations.
It is said that the system does
Free oscillationsIf it is removed once from the position of the equilibrium and in the future provided by herself. No periodic external
The impacts of the system does not have any internal energy sources that support oscillations in the system.
The fluctuations in spring and mathematical pendulum discussed above are examples of free oscillations.
The frequency with which free oscillations are performed is called own frequency oscillatory system. So, formulas (9) и (12) They give their own (cyclic) frequencies of springs and mathematical pendulums.
In an idealized situation in the absence of friction, free oscillations are unsuccessful, i.e., they have a permanent amplitude and lasts indefinitely. In real oscillatory systems, friction is always present, so free oscillations are gradually faded (Fig. 6).
Fig. 6. Flowering oscillations |
Forced oscillations - these are oscillations performed by the system under the influence of external force , periodically changing in time (the so-called forcing force).
Suppose your own frequency of system oscillations is equal , and the generating force depends on the time of harmonic law:
.
For some time, forced oscillations are established: the system makes a complex movement, which is the imposition of uniformed and free oscillations. Free oscillations are gradually faded, and in the steady mode, the system performs forced oscillations, which also turn out to be harmonious. The frequency of established forced oscillations coincides with the frequency forgoing power (external force as if impose a system of its frequency).
The amplitude of the established forced oscillations depends on the frequency of the forcing force. The graph of this dependence is shown in Fig. 7.
Fig. 7. Resonance |
We see that near the frequency There is a resonance - a phenomenon of increasing the amplitude of forced oscillations. The resonant frequency is approximately equal to the system of system oscillations: , And this equality is done more precisely, the less friction in the system. In the absence of friction, the resonant frequency coincides with its own oscillation frequency, , and the amplitude of oscillations increases indefinitely .
The amplitude of oscillations is the maximum value of the deviation from the zero point. In physics, this process is analyzed in different sections.
It is studied with mechanical, sound and electromagnetic oscillations. In listed cases, the amplitude is measured differently and in its laws.
Oscillation amplitude
The amplitude of oscillations call the maximum remote point of finding the body from the equilibrium position. In physics, it is indicated by the letter A and measured in meters.
The amplitude can be observed on a simple example of a spring pendulum.
In the perfect case, when the resistance of the airspace and the friction of the spring device is ignored, the device will fluctuate infinitely. Motion Description is performed using COS and SIN functions:
x (t) = a * cos (ωt + φ0) or x (t) = a * sin (ωt + φ0),
Where
The value A is the amplitude of the free movements of the cargo on the spring;
(ωt + φ0) is the phase of free oscillations, where ω is a cyclic frequency, and φ0 is the initial phase when T = 0.
In physics, the specified formula is called the equation of harmonic oscillations. This equation fully discloses a process where the pendulum moves with a certain amplitude, period and frequency.
Period of oscillations
The results of laboratory experiments show that the cyclic period of cargo movement on the spring directly depends on the mass of the pendulum and the rigidity of the spring, but does not depend on the amplitude of the movement.
In physics, the period is denoted by the letter T and describe with formulas:
Based on the formula, the period of oscillations are mechanical movements that are repeated after a certain period of time. Simple words, the period is called one complete movement of cargo.
Frequency of oscillations
Under the frequency of oscillations, it is necessary to understand the number of repetitions of the movement of the pendulum or the passage of the wave. In different sections of physics, the frequency is indicated by letters ν, f or f.
This value is described by the expression:
V = N / T - the number of oscillations over time
Where
In the international measurement system, the frequency is measured in Hz (Hertz). It refers to the exact measured component of the oscillatory process.
For example, the science is installed the frequency of the sun around the center of the universe. It is - 10. ^{35. }Hz at the same speed.
Cyclic frequency
In physics, cyclic and circular frequency have the same value. This value is also called an angular frequency.
Denote her letter Omega. It is equal to the number of its own oscillatory movements of the body for 2π seconds of time:
Ω = 2π / T = 2πν.
This value found its use in radio engineering and, based on mathematical calculation, has a scalar characteristic. Its measurements are carried out in radians for a second. With its help, the calculations of processes in radio engineering are greatly simplified.
For example, the resonant value of the angular frequency of the oscillating circuit is calculated by the formula:
WLC = 1 / LC.
Then the usual cyclic resonance frequency is expressed:
VLC = 1/2 2π * √ LC.
In the electrician under the angular frequency, it is necessary to understand the number of emf transformations or the number of radius revolutions - vector. Here it is denoted by the letter f.
How to determine the amplitude, period and frequency of fluctuations on schedule
To determine the components of the components of the oscillatory mechanical process or, for example, fluctuations in temperature, you need to understand the terms of this process.
These include:
The distance of the test object from the original point is called displacement and denotes x;
The greatest deviation is the amplitude of the displacement A;
oscillation phase - determines the state of the oscillating system at any time;
The initial phase of the oscillatory process - when T = 0, then φ = φ _{0}.
From the graph, it can be seen that the value of the sinus and cosine can vary from -1 to +1. So, the displacement x can be equal to-and + a. Movement from -a to + and is called a complete oscillation.
The built schedule clearly shows the period and frequency of oscillations. It should be noted that the phase does not affect the shape of the curve, and only affects its position at a given period of time.