The Coriolis force is:
where - point weight,-vectorangular velocity of a rotating frame of reference, is the velocity vector of a point mass in this frame of reference, square brackets indicate the operation vector product.
Value 
called Coriolis acceleration.
By physical nature
Mechanical(sound,vibration)
electromagnetic (light,radio waves, thermal)
mixed type- combinations of the above
By the nature of interaction with the environment
Forced - fluctuations occurring in the system under the influence of external periodic influence. Examples: leaves on trees, raising and lowering a hand. With forced vibrations, a phenomenon may occur resonance: a sharp increase in the amplitude of oscillations at the coincidence natural frequencyoscillator and frequency of external influence.
Free (or own)- these are oscillations in the system under the action of internal forces, after the system is taken out of equilibrium (in real conditions, free oscillations are always fading). The simplest examples of free vibrations are the vibrations of a load attached to a spring, or a load suspended from a thread.
Self-oscillations - fluctuations in which the system has a margin potential energy, spent on making oscillations (an example of such a system is mechanical watches). A characteristic difference between self-oscillations and forced oscillations is that their amplitude is determined by the properties of the system itself, and not by the initial conditions.
Parametric - fluctuations that occur when any parameter of the oscillatory system changes as a result of external influence.
Random - fluctuations in which the external or parametric load is a random process.
Harmonic vibrations
where XBUTω
Generalized harmonic oscillation in differential form
(Any non-trivial
Velocity and acceleration in harmonic oscillations.
According to the definition of speed, speed is the derivative of the coordinate with respect to time
Thus, we see that the speed during harmonic oscillatory motion also changes according to the harmonic law, but the speed fluctuations are ahead of the displacement fluctuations in phase by p/2.
The value is the maximum speed of oscillatory motion (amplitude of speed fluctuations).
Therefore, for the speed during harmonic oscillation we have: 
,
and for the case of a zero initial phase (see graph).
According to the definition of acceleration, acceleration is the derivative of speed with respect to time:
-
the second derivative of the coordinate with respect to time. Then: .
Acceleration during harmonic oscillatory motion also changes according to a harmonic law, but acceleration oscillations are ahead of velocity oscillations by p / 2 and displacement oscillations by p (they say that oscillations occur out of phase).
Value
Maximum acceleration (amplitude of acceleration fluctuations). Therefore, for acceleration we have: 
,
and for the case of zero initial phase: 
(see graph).
From the analysis of the process of oscillatory motion, graphs and corresponding mathematical expressions, it can be seen that when the oscillating body passes the equilibrium position (displacement is zero), the acceleration is zero, and the body speed is maximum (the body passes the equilibrium position by inertia), and when the amplitude value of the displacement is reached, the speed is equal to zero, and the acceleration is maximal in absolute value (the body changes the direction of its motion).
Harmonic vibrations- fluctuations in which a physical (or any other) quantity changes over time according to a sinusoidal or cosine law. The kinematic equation of harmonic oscillations has the form
where X- displacement (deviation) of the oscillating point from the equilibrium position at time t; BUT- oscillation amplitude, this is the value that determines the maximum deviation of the oscillating point from the equilibrium position; ω - cyclic frequency, a value showing the number of complete oscillations occurring within 2π seconds; - full phase of oscillations; - initial phase of oscillations.
Generalized harmonic oscillation in differential form
(Any non-trivial the solution of this differential equation is a harmonic oscillation with a cyclic frequency)
(lat. amplitude- magnitude) - this is the largest deviation of the oscillating body from the equilibrium position.
For a pendulum, this is the maximum distance that the ball moves from its equilibrium position (figure below). For oscillations with small amplitudes, this distance can be taken as the length of the arc 01 or 02, as well as the lengths of these segments.
The oscillation amplitude is measured in units of length - meters, centimeters, etc. On the oscillation graph, the amplitude is defined as the maximum (modulo) ordinate of the sinusoidal curve, (see figure below).
Oscillation period.
Oscillation period- this is the smallest period of time after which the system, making oscillations, again returns to the same state in which it was at the initial moment of time, chosen arbitrarily.
In other words, the oscillation period ( T) is the time for which one complete oscillation takes place. For example, in the figure below, this is the time it takes for the weight of the pendulum to move from the rightmost point through the equilibrium point O to the leftmost point and back through the point O again to the far right.
For a full period of oscillation, therefore, the body travels a path equal to four amplitudes. The oscillation period is measured in units of time - seconds, minutes, etc. The oscillation period can be determined from the well-known oscillation graph, (see figure below).
The concept of “oscillation period”, strictly speaking, is valid only when the values of the oscillating quantity are exactly repeated after a certain period of time, that is, for harmonic oscillations. However, this concept is also applied to cases of approximately repeating quantities, for example, for damped oscillations.
Oscillation frequency.
Oscillation frequency is the number of oscillations per unit of time, for example, in 1 s.
The SI unit of frequency is named hertz(Hz) in honor of the German physicist G. Hertz (1857-1894). If the oscillation frequency ( v) is equal to 1 Hz, then this means that one oscillation is made for every second. The frequency and period of oscillations are related by the relations:
In the theory of oscillations, the concept is also used cyclical, or circular frequency ω . It is related to the normal frequency v and oscillation period T ratios:
.
Cyclic frequency is the number of oscillations per 2π seconds.
So far, we have considered natural oscillations, i.e., oscillations that occur in the absence of external influences. External influence was needed only to bring the system out of equilibrium, after which it was left to itself. The differential equation of natural oscillations does not contain any traces of external influence on the system at all: this influence is reflected only in the initial conditions.
Establishment of vibrations. But very often one has to deal with fluctuations that occur with a constantly present external influence. Especially important and at the same time quite simple to study is the case when the external force has a periodic character. A common feature of forced oscillations occurring under the action of a periodic external force is that some time after the onset of the external force, the system completely “forgets” its initial state, the oscillations become stationary and do not depend on the initial conditions. The initial conditions appear only during the period of oscillation establishment, which is usually called the transition process.
sinusoidal effect. Let us first consider the simplest case of forced oscillations of an oscillator under the action of an external force that changes according to a sinusoidal law:
Rice. 178. Excitation of forced oscillations of a pendulum
Such an external influence on the system can be carried out in various ways. For example, you can take a pendulum in the form of a ball on a long rod and a long spring with low stiffness and attach it to the pendulum rod near the suspension point, as shown in Fig. 178. The other end of a horizontal spring should be forced to move according to the law? using a crank mechanism driven by an electric motor. Current
on the pendulum from the side of the spring, the driving force will be practically sinusoidal if the range of motion of the left end of the spring B will be much greater than the amplitude of the oscillations of the pendulum rod at the point of attachment of the spring C.
The equation of motion. The equation of motion for this and other similar systems, in which, along with the restoring force and the resistance force, the oscillator is acted upon by an external driving force that varies sinusoidally with time, can be written as
Here the left side, in accordance with Newton's second law, is the product of mass and acceleration. The first term on the right side is the restoring force proportional to the displacement from the equilibrium position. For a load suspended on a spring, this is an elastic force, and in all other cases, when its physical nature is different, this force is called quasi-elastic. The second term is the friction force proportional to the speed, for example, the air resistance force or the friction force in the axis. The amplitude and frequency of the driving force swinging the system will be considered constant.
We divide both sides of equation (2) by the mass and introduce the notation
Now equation (2) takes the form
In the absence of a driving force, the right side of Eq. (4) vanishes and, as expected, it reduces to the equation of natural damped oscillations.
Experience shows that in all systems, under the action of a sinusoidal external force, oscillations are eventually established, which also occur according to the sinusoidal law with the frequency of the driving force co and with a constant amplitude a, but with some phase shift relative to the driving force. Such oscillations are called steady-state forced oscillations.
Steady fluctuations. Let us first consider the steady-state forced oscillations, and for simplicity, we will neglect friction. In this case, there will be no velocity term in equation (4):
Let's try to look for a solution corresponding to steady-state forced oscillations, in the form
Let us calculate the second derivative and substitute it together with into equation (5):
For this equality to be valid at any time, the coefficients for the left and right must be the same. From this condition we find the oscillation amplitude a:
Let us investigate the dependence of the amplitude a on the frequency of the driving force. The graph of this dependence is shown in Fig. 179. At , formula (8) gives. Substituting the values here, we see that the time-constant force simply shifts the oscillator to a new equilibrium position shifted from the old one by From (6) it follows that at the displacement
as it obviously should be.
Rice. 179. Addiction Graph
Phase ratios. As the frequency of the driving force increases from 0 to, steady-state oscillations occur in phase with the driving force, and their amplitude constantly increases, slowly at first, and as it approaches to, faster and faster: at , the oscillation amplitude increases indefinitely
For values \u200b\u200bof exceeding the frequency of natural vibrations, formula (8) gives a negative value for a (Fig. 179). It is clear from formula (6) that at , oscillations occur in antiphase with the driving force: when the force acts in one direction, the oscillator is shifted in the opposite direction. With an unlimited increase in the frequency of the driving force, the oscillation amplitude tends to zero.
In all cases, it is convenient to consider the amplitude of oscillations to be positive, which can be easily achieved by introducing a phase shift between the driving
force and displacement:
Here, a is still given by formula (8), and the phase shift is zero at and is equal to at. 180.
Rice. 180. Amplitude and phase of forced oscillations
Resonance. The dependence of the amplitude of forced oscillations on the frequency of the driving force is nonmonotonic. A sharp increase in the amplitude of forced oscillations as the frequency of the driving force approaches the natural frequency of the oscillator is called resonance.
Formula (8) gives an expression for the amplitude of forced oscillations neglecting friction. It is precisely with this neglect that the oscillation amplitude turns to infinity with exact coincidence of frequencies. In reality, the oscillation amplitude cannot, of course, turn to infinity.
This means that when describing forced oscillations near resonance, it is fundamentally necessary to take friction into account. When friction is taken into account, the amplitude of forced oscillations at resonance is finite. It will be the smaller, the greater the friction in the system. Far from resonance, formula (8) can be used to find the oscillation amplitude even in the presence of friction, if it is not too strong, i.e. Moreover, this formula, obtained without friction, has physical meaning only when friction is still present . The fact is that the very concept of steady forced oscillations is applicable only to systems in which there is friction.
If there were no friction at all, then the process of establishing oscillations would continue indefinitely. In reality, this means that expression (8) for the amplitude of forced oscillations, obtained without taking into account friction, will correctly describe oscillations in the system only after a sufficiently long period of time after the start of the driving force. The words "sufficiently large period of time" mean here that the transient process has already ended, the duration of which coincides with the characteristic decay time of natural oscillations in the system.
At low friction, steady forced oscillations occur in phase with the driving force at and in antiphase at as well as in the absence of friction. However, near resonance, the phase does not change abruptly, but continuously, and with an exact coincidence of frequencies, the displacement lags behind the driving force in phase by (a quarter of the period). The speed changes in this case in phase with the driving force, which provides the most favorable conditions for the transfer of energy from the source of the external driving force to the oscillator.
What is the physical meaning of each of the terms in equation (4), which describes the forced oscillations of an oscillator?
What is a steady state forced oscillation?
Under what conditions can formula (8) be used for the amplitude of steady-state forced oscillations obtained without friction?
What is resonance? Give examples of the manifestation and use of the phenomenon of resonance known to you.
Describe the phase shift between the driving force and the displacement at different ratios between the frequency co in the driving force and the natural frequency of the oscillator.
What determines the duration of the process of establishing forced oscillations? Give a rationale for your answer.
Vector diagrams. You can verify the validity of the above statements if you obtain a solution to equation (4), which describes the steady forced oscillations in the presence of friction. Since steady oscillations occur with the driving force frequency ω and some phase shift, the solution of Eq. (4) corresponding to such oscillations should be sought in the form
In this case, the speed and acceleration, obviously, will also change with time according to the harmonic law:
It is convenient to determine the amplitude a of steady forced oscillations and the phase shift using vector diagrams. Let us take advantage of the fact that the instantaneous value of any quantity changing according to the harmonic law can be represented as a projection of a vector onto some pre-selected direction, and the vector itself rotates uniformly in a plane with a frequency ω, and its constant length is equal to
amplitude value of this oscillating quantity. In accordance with this, we compare each term of equation (4) with a vector rotating with angular velocity, the length of which is equal to the amplitude value of this term.
Since the projection of the sum of several vectors is equal to the sum of the projections of these vectors, equation (4) means that the sum of the vectors associated with the terms on the left side is equal to the vector associated with the value on the right side. To construct these vectors, we write out the instantaneous values of all terms on the left side of equation (4), taking into account the relations
It can be seen from formulas (13) that the length vector associated with the value is ahead by an angle of the vector associated with the value The length vector associated with the term x is ahead by the length vector, i.e. these vectors are directed in opposite directions.
The mutual arrangement of these vectors for an arbitrary moment of time is shown in fig. 181. The whole system of vectors rotates as a whole with an angular velocity co counterclockwise around the point O.
Rice. 181. Vector diagram of forced oscillations
Rice. 182. Vector associated with an external force
The instantaneous values of all quantities are obtained by projecting the corresponding vectors onto a preselected direction. The vector associated with the right side of equation (4) is equal to the sum of the vectors shown in fig. 181. This addition is shown in fig. 182. Applying the Pythagorean theorem, we obtain
whence we find the amplitude of steady-state forced oscillations a:
The phase shift between driving force and displacement as seen from the vector diagram in fig. 182 is negative because the length vector lags behind the vector Therefore
So, steady forced oscillations occur according to the harmonic law (10), where a and are determined by formulas (14) and (15).
Rice. 183. Dependence of the amplitude of forced oscillations on the frequency of the driving force
resonance curves. The amplitude of steady-state forced oscillations is proportional to the amplitude of the driving force Let us study the dependence of the oscillation amplitude on the frequency of the driving force. At low damping y, this dependence is very sharp. If then, as ω tends to the frequency of free oscillations, the amplitude of forced oscillations a tends to infinity, which coincides with the previously obtained result (8). In the presence of damping, the oscillation amplitude at resonance no longer goes to infinity, although it significantly exceeds the oscillation amplitude under the action of an external force of the same magnitude, but having a frequency far from the resonant one. Resonance curves for different values of the damping constant y are shown in Figs. 183. To find the resonance frequency of cores, it is necessary to find at what co the radical expression in formula (14) has a minimum. Equating the derivative of this expression with respect to zero (or supplementing it to a full square), we make sure that the maximum amplitude of forced oscillations occurs at
The resonant frequency turns out to be less than the frequency of free oscillations of the system. For small 7, the resonant frequency practically coincides with As the frequency of the driving force tends to infinity, i.e., at , the amplitude a, as can be seen from (14), tends to zero. With that is, under the action of a constant external force, the amplitude If we substitute here and get This is the static displacement of the oscillator from the equilibrium position under the action of a constant force and the displacement of the oscillator occurs in antiphase with the driving force. At resonance, as can be seen from (15), the displacement lags behind the external force in phase by The second of formulas (13) shows that in this case, the external force changes in phase with velocity, i.e., it acts in the direction of motion all the time. That this is exactly how it should be is clear from intuitive considerations.
Resonance of speed. It can be seen from formula (13) that the amplitude of the speed oscillations at steady state forced oscillations is equal to . Using (14), we obtain
Rice. 184. Velocity amplitude at steady forced oscillations
The dependence of the velocity amplitude on the frequency of the external force is shown in fig. 184. The resonance curve for velocity, although similar to the resonance curve for displacement, differs from it in some respects. So, when, i.e., under the action of a constant force, the oscillator experiences a static displacement from the equilibrium position and its speed after the transition process ends is equal to zero. It can be seen from formula (19) that the velocity amplitude at vanishes. Velocity resonance occurs when the frequency of the external force coincides exactly with the frequency of free oscillations
How are vector diagrams constructed for steady forced oscillations under a sinusoidal external action?
What determines the frequency, amplitude and phase of steady forced harmonic oscillations?
Describe the differences between the resonance curves for displacement amplitude and velocity amplitude. What characteristics of the oscillatory system determine the sharpness of the resonance curves?
How is the nature of the resonance curve related to the parameters of the system that determine the damping of its own oscillations?
