THE HARMONIOUS OSCILLATIVE MOTION

§1 Kinematics of harmonious oscillations

 

Processes, repeated over time are called vibrations..

Depending on the nature of vibrational process and the excitation mechanism  are: mechanical oscillations (oscillations of pendulums, strings, buildings, a ground surface etc.); electromagnetic oscillations (alternating current oscillations, oscillations of vectors and and  in an electromagnetic wave etc.); electromechanical oscillations (oscillations of a membrane of phone, a loudspeaker diaphragm, etc.); vibrations of the nuclei  and molecules as a result of a thermal motion in atoms.

Consider the segment [OD] (the radius vector), perform a rotational movement around the point 0. Length | AP | = A. Rotation occurs at a constant angular velocity ω0. Then the angle φ between the radius vector and the axis of x varies with time as

where φ0 - angle between [OD] and the x-axis at time t = 0. Projection of [OD] on the x-axis at time t = 0

and at any given time

 (1)

Thus, the projection of [OD] on the x-axis oscillates occurring along the x axis, and these fluctuations are described by cosine law (Eq. (1)).

Vibrations, which are described by the cosine

or a sine  

called harmonic.

Harmonic oscillations are periodic, as value of x(y) is repeated at regular intervals.

If [OD] s is the lowest position in the picture, ie Point D is the point P, then its projection on the x-axis is zero. We call this state of [OD] equilibrium position. Then we can say that the value of x describes the displacement of the vibrating point of equilibrium. The maximum displacement from the equilibrium position is called the amplitude fluctuations

Quantity

which is under the sign of the cosine of the phase is called. Phase determines the displacement from equilibrium at an arbitrary time t. Phase at the initial time t = 0, equal to φ0 is called the initial phase.

 

 

The length of time for which is made one complete oscillation is called the oscillation period T. The number of oscillations per unit of time is called the oscillation frequency ν

 

After a time interval equal to the period T, ie, an increase in the argument of the cosine ω0T, movement is repeated, and the cosine takes the old value

Since the period of cosine is equal 2 π, hence, ω0 = 2π

Figure harmonic oscillation

  A - amplitude, period, х - displacement, t – time.

 

thus, ω0 - is the number of oscillations of the body for the 2π seconds. ω0 - circular or angular frequency.

Speed ??oscillating point is found by differentiating equation displacement x(t) in time

 

ie speed v is different in phase from the displacement x on π/2.

Acceleration - the first derivative of the velocity (the second derivative of the displacement) over time

 

ie acceleration a different from displacement in phase on π.


Plot х (t), v (t) and a (t)in the same coordinate system (for simplicity we take φ0 = 0 and ω0 = 1) .

Free or own called vibrations that occur in the system to its own after it was removed from the equilibrium position.

 

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