| These are the motions of particles (bodies) possessing repetition in time by any extent. |
Period of oscillations (Т) | This is the time of one complete oscillation.
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Frequency of oscillations (n) | This is the number (N) of oscillations that are completed in a unit time. The SI unit is the hertz (Hz): one oscillation per second. |
Dependence between n and Т | |
| This is the number of oscillations made in 2p second. The SI unit is s-1. |
Connection between the angular frequency (w), the frequency (ν) and the period of oscillations (T) | |
Amplitude of oscillations (А or хm). | This is the maximum displacement of the body from the equilibrium position. |
| These are the oscillations which result in the observed quantity (displacement) (х) varies with time according to sinusoidal or cosine function : х = А×cos(w×t + jo), A is the amplitude; w is the angular frequency of oscillations; (w×t + jo) is the phase of oscillations; jo is the initial phase (at t = 0). |
Free or natural oscillations | The oscillations which occur in the system left by its own after being disturbed from the equilibrium position. |
Conditions of an occurrence of free harmonic oscillations | The frictional forces are absent in an oscillated system. The elastic force occurs at the deflection of a body from the equilibrium position. |
| The force is due to the deflection of a body from the equilibrium position, it is proportional to the displacement and directed to the equilibrium position (opposite to the displacement): Fel = -k×x, k is the elasticity coefficient. |
Natural frequency (wо) of free harmonic oscillations (calculation formula) | wо = Ök/m, k is the elasticity coefficient; m is the mass of a body. |
Period of oscillations of a mathematical (simple) pendulum (T) | Т = 2pÖL/g L is the length of a string; g is the free fall acceleration. |
| These are free oscillations with availability of drag forces (the frictional force). |
Dependence of a drag force on the velocity of a body motion | Fd = -r×v = -r×dx/dt, r is the drag coefficient dependent on the property of the medium, the form and sizes of the body; v is the velocity of the motion. |
| b = r/2m, r is the drag coefficient; m is the body mass. |
Condition of an occurrence of damped oscillations | b < wо (periodic damping), wо is the angular frequency of free oscillations at the absence of the friction. |
Law descriptive of damped oscillations | x = Aо×exp(-b×t)×cos(wd×t), Ao is the initial amplitude; b is the damping constant; wd is the angular frequency of damped oscillations. |
Amplitude (A) and the frequency of damped oscillations (wd) | А = Aо×exp(-b×t), wd = Ö(wo2 - b2) |
Logarithmic decrement of damping (λ) | This is the natural logarithm of the ratio of two successive amplitudes separated by the period of oscillations. l = ln[A(t)/A(t + T)] = ln[Ai/Ai+1], i is the serial number of the oscillation. |
Calculation formula for a logarithmic decrement (damping constant) | l= b×T, b is the damping constant; T is the period of damped oscillations. |
| These are the oscillations at which the oscillated system is subjected to the action of an external periodical force (it is termed a driving force). |
Harmonic driving force (Fdr) | Fdr= Fо×cos(wdr×t), Fo is the amplitude of a driving force; ωdr is the angular frequency of a driving force. |
Frequency of forced oscillations | It is equal to the frequency of a driving force (ωdr). |
Amplitude of forced oscillations (A) | |
| This is the achievement of the maximum (greatest) amplitude of forced oscillations at the certain value of the driving force frequency. |
Frequency of a driving force (ωres) at which the resonance occurs. | |
Amplitude at the resonance (Ares) | |
Total energy of simple (E) harmonic oscillations | E = k×А2/2 = m×А2×w2/2, m is the body mass, А is the amplitude, w is the angular frequency. |