Lecture 8. Vibrations

Small deviations from equilibrium

Harmonic oscillation is the periodic variation of a quantity, which can be expressed as a sine or cosine function. The following classifications are used. On the basis of proceeding from the nature of oscillations: mechanical and electromagnetic. On the basis of proceeding from external forces: free and forced oscillation. On the basis of proceeding of the energy changing: undamped and damped oscillations. On the basis of proceeding from the form of equations: harmonic and unharmonic oscillations.

Characteristics of harmonic oscillations:

- the period T is the time of one full oscillation;

- the frequency n is the number of oscillations per unit of time: , [n]=Hz. - the cyclic frequency:

, [w₀] = . (8.1)

Any harmonic oscillation can be illustrated by the graph or by the vector diagram. Fig.8.1 represents graph of function

. (8.2)

Figure 8.1

For vector diagrams complex numbers are used. Any complex number contains Value i is called imaginary unit. Now let’s construct the vector diagram of equation

; (8.3)

; .

The vector is of 2 m length and is situated under the initial angle . The vector is rotation of the watch hand with angle velocity .

Figure 8.2

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