The simplest type of vibrations are harmonic vibrations- fluctuations in which the displacement of the oscillating point from the equilibrium position changes over time according to the sine or cosine law.
So, with a uniform rotation of the ball around the circumference, its projection (shadow in parallel rays of light) performs a harmonic oscillatory motion on a vertical screen (Fig. 1).
The displacement from the equilibrium position during harmonic vibrations is described by the equation (it is called the kinematic law harmonic movement) of the form:
where x - displacement - a value that characterizes the position of the oscillating point at time t relative to the equilibrium position and is measured by the distance from the equilibrium position to the position of the point in this moment time; A - oscillation amplitude - the maximum displacement of the body from the equilibrium position; T - oscillation period - the time of one complete oscillation; those. the smallest period of time after which the values of physical quantities characterizing the oscillation are repeated; - initial phase;
The phase of the oscillation at time t. The oscillation phase is the argument periodic function, which at a given oscillation amplitude determines the state of the oscillatory system (displacement, speed, acceleration) of the body at any time.
If in initial moment time, the oscillating point is maximally displaced from the equilibrium position, then , and the displacement of the point from the equilibrium position changes according to the law
If the oscillating point at is in a position of stable equilibrium, then the displacement of the point from the equilibrium position changes according to the law
The value of V, the reciprocal of the period and equal to the number full oscillations made in 1 s is called the frequency of oscillations:
If in time t the body makes N complete oscillations, then
the value 
, showing how many oscillations the body makes in s, is called cyclic (circular) frequency.
The kinematic law of harmonic motion can be written as:
Graphically, the dependence of the displacement of an oscillating point on time is represented by a cosine (or sinusoid).
Figure 2, a shows the time dependence of the displacement of the oscillating point from the equilibrium position for the case .
Let us find out how the speed of an oscillating point changes with time. To do this, we find the time derivative of this expression:
where is the amplitude of the velocity projection on the x-axis.
This formula shows that during harmonic oscillations, the projection of the body velocity on the x axis also changes according to the harmonic law with the same frequency, with a different amplitude, and is ahead of the mixing phase by (Fig. 2, b).
To find out the dependence of acceleration, we find the time derivative of the velocity projection:
where is the amplitude of the acceleration projection on the x-axis.
For harmonic oscillations, the acceleration projection leads the phase shift by k (Fig. 2, c).
Similarly, you can build dependency graphs
Considering that , the formula for acceleration can be written
those. for harmonic oscillations, the acceleration projection is directly proportional to the displacement and opposite in sign, i.e. acceleration is directed in the direction opposite to the displacement.
So, the acceleration projection is the second derivative of the displacement, then the resulting ratio can be written as:
The last equality is called equation of harmonic oscillations.
A physical system in which harmonic oscillations can exist is called harmonic oscillator, and the equation of harmonic oscillations - harmonic oscillator equation.
Harmonic Wave Equation
The harmonic oscillation equation establishes the dependence of the body coordinate on time
The cosine graph has a maximum value at the initial moment, and the sine graph has a zero value at the initial moment. If we begin to investigate the oscillation from the equilibrium position, then the oscillation will repeat the sinusoid. If we begin to consider the oscillation from the position of the maximum deviation, then the oscillation will describe the cosine. Or such an oscillation can be described by the sine formula with an initial phase.
Change in speed and acceleration during harmonic oscillation
Not only the coordinate of the body changes with time according to the law of sine or cosine. But such quantities as force, speed and acceleration also change in a similar way. Force and acceleration are maximum when the oscillating body is in extreme positions, where the displacement is maximum, and are equal to zero when the body passes through the equilibrium position. The speed, on the contrary, in the extreme positions is equal to zero, and when the body passes the equilibrium position, it reaches its maximum value.
If the oscillation is described according to the law of cosine
If the oscillation is described according to the sine law
Maximum speed and acceleration values
After analyzing the equations of dependence v(t) and a(t), one can guess that the maximum values of velocity and acceleration take on when trigonometric factor is 1 or -1. Determined by the formula
Changes in time according to a sinusoidal law:
where X- the value of the fluctuating quantity at the moment of time t, BUT- amplitude , ω - circular frequency, φ is the initial phase of oscillations, ( φt + φ ) is the total phase of oscillations . At the same time, the values BUT, ω and φ - permanent.
For mechanical vibrations with an oscillating value X are, in particular, displacement and velocity, for electrical oscillations- voltage and current.
Harmonic vibrations take special place among all types of oscillations, since this is the only type of oscillations, the shape of which is not distorted when passing through any homogeneous environment, i.e., waves propagating from a source of harmonic oscillations will also be harmonic. Any non-harmonic vibration can be represented as a sum (integral) of various harmonic vibrations (in the form of a spectrum of harmonic vibrations).
Energy transformations during harmonic vibrations.
In the process of oscillations, there is a transition of potential energy Wp into kinetic W k and vice versa. In the position of maximum deviation from the equilibrium position, the potential energy is maximum, the kinetic energy is zero. As you return to the equilibrium position, the speed of the oscillating body increases, and with it, so does kinetic energy, reaching a maximum in the equilibrium position. The potential energy then drops to zero. Further-neck movement occurs with a decrease in speed, which drops to zero when the deflection reaches its second maximum. Potential energy here increases to its initial (maximum) value (in the absence of friction). Thus, the oscillations of the kinetic and potential energies occur with a double (compared to the oscillations of the pendulum itself) frequency and are in antiphase (i.e., there is a phase shift between them equal to π ). Total vibration energy W remains unchanged. For a body oscillating under the action of an elastic force, it is equal to:
where v m- the maximum speed of the body (in the equilibrium position), x m = BUT- amplitude.
Due to the presence of friction and resistance of the medium free vibrations decay: their energy and amplitude decrease with time. Therefore, in practice, not free, but forced oscillations are used more often.
Along with progressive and rotational movements bodies in mechanics, oscillatory motions are also of considerable interest. Mechanical vibrations called the movements of bodies that repeat exactly (or approximately) at regular intervals. The law of motion of an oscillating body is given by some periodic function of time x = f (t). Graphic image This function gives a visual representation of the course of the oscillatory process in time.
Examples of simple oscillatory systems are a load on a spring or mathematical pendulum(Fig. 2.1.1).
Mechanical vibrations, like oscillatory processes any other physical nature, may be free and forced. Free vibrations are made under the influence internal forces system after the system has been brought out of equilibrium. The oscillations of a weight on a spring or the oscillations of a pendulum are free oscillations. vibrations under the action external periodically changing forces are called forced .
The simplest type of oscillatory process are simple harmonic vibrations , which are described by the equation
|
x = x m cos (ω t + φ 0). |
Here x- displacement of the body from the equilibrium position, x m - oscillation amplitude, i.e. the maximum displacement from the equilibrium position, ω - cyclic or circular frequency hesitation, t- time. The value under the cosine sign φ = ω t+ φ 0 is called phase harmonic process. At t= 0 φ = φ 0 , so φ 0 is called initial phase. The minimum time interval after which the movement of the body is repeated is called period of oscillation T. Physical quantity, the reciprocal of the oscillation period, is called oscillation frequency:
Oscillation frequency f shows how many vibrations are made in 1 s. Frequency unit - hertz(Hz). Oscillation frequency f is related to the cyclic frequency ω and the oscillation period T ratios:
On fig. 2.1.2 shows the positions of the body at regular intervals with harmonic vibrations. Such a picture can be obtained experimentally by illuminating an oscillating body with short periodic flashes of light ( stroboscopic lighting). The arrows represent the velocity vectors of the body at different points in time.
Rice. 2.1.3 illustrates the changes that occur on the graph of a harmonic process if either the amplitude of the oscillations changes x m , or period T(or frequency f), or the initial phase φ 0 .
At oscillatory motion bodies along a straight line (axis OX) the velocity vector is always directed along this straight line. Velocity υ = υ x body movement is determined by the expression
In mathematics, the procedure for finding the limit of the ratio at Δ t→ 0 is called the calculation of the derivative of the function x (t) by time t and denoted as or as x"(t) or finally as . For the harmonic law of motion Calculation of the derivative leads to the following result:
The appearance of the term + π / 2 in the cosine argument means a change in the initial phase. Maximum modulo values of velocity υ = ω x m are achieved at those moments of time when the body passes through the equilibrium positions ( x= 0). Acceleration is defined in a similar way a = ax bodies with harmonic vibrations:
hence the acceleration a is equal to the derivative of the function υ ( t) by time t, or the second derivative of the function x (t). The calculations give:
The minus sign in this expression means that the acceleration a (t) always has a sign, opposite sign bias x (t), and, therefore, according to Newton's second law, the force that causes the body to perform harmonic oscillations is always directed towards the equilibrium position ( x = 0).
fluctuations called movements or processes that are characterized by a certain repetition in time. Oscillatory processes are widespread in nature and technology, for example, the swing of a clock pendulum, variable electricity etc. When the pendulum oscillates, the coordinate of its center of mass changes, in the case alternating current voltage and current fluctuate in the circuit. physical nature oscillations can be different, therefore, mechanical, electromagnetic, etc. oscillations are distinguished. However, various oscillatory processes are described by the same characteristics and the same equations. From this comes the feasibility unified approach to the study of vibrations different physical nature.
The fluctuations are called free, if they are performed only under the influence of internal forces acting between the elements of the system, after the system is removed from the equilibrium position external forces and left to itself. Free vibrations always damped oscillations , because in real systems energy losses are inevitable. In the idealized case of a system without energy loss, free oscillations (continuing as long as desired) are called own.
The simplest type of free undamped oscillations are harmonic oscillations - fluctuations in which the fluctuating value changes with time according to the sine (cosine) law. Oscillations encountered in nature and technology often have a character close to harmonic.
Harmonic vibrations are described by an equation called the equation of harmonic vibrations:
where BUT- amplitude of fluctuations, the maximum value of the fluctuating value X; - circular (cyclic) frequency of natural oscillations; - the initial phase of the oscillation at a moment of time t= 0; - the phase of the oscillation at the moment of time t. The phase of the oscillation determines the value of the oscillating quantity at a given time. Since the cosine varies from +1 to -1, then X can take values from + A before - BUT.
Time T, for which the system completes one complete oscillation, is called period of oscillation. During T oscillation phase is incremented by 2 π , i.e.
Where . (14.2)
The reciprocal of the oscillation period
i.e., the number of complete oscillations per unit time is called the oscillation frequency. Comparing (14.2) and (14.3) we obtain
The unit of frequency is hertz (Hz): 1 Hz is the frequency at which one complete oscillation takes place in 1 s.
Systems in which free vibrations can occur are called oscillators . What properties must a system have in order for free oscillations to occur in it? mechanical system must have position of stable equilibrium, upon exiting which appears restoring force towards equilibrium. This position corresponds, as is known, to a minimum potential energy systems. Let us consider several oscillatory systems that satisfy the listed properties.
