Kinematics of Harmonic Oscillations

From the entire variety of periodic oscillations we will select first of all the so-called harmonic oscillations. Interest in harmonic oscillations is due to the following reasons firstly, it is relatively simple to describe harmonic oscillations mathematically, and, secondly, any periodic oscillations can be presented as a superposition of harmonic oscillations. This latter circumstance is very important, and we will return to it in Section 2.3.2. 

Harmonic oscillations are an abstraction, since they have to continue for an infinitely long period (- f + ), according to certain laws, without any changes, which is not 

Oscillations are referred to as harmonic if the changes in time of some physical values occur under the sine or cosine law 

The choice of sine or cosine for describing the harmonic oscillations as well as the initial phase is rather arbitrary and is chosen for convenience. A cosine form is preferable in most cases as will be seen later. The transition from one form to another is easily realized by a corresponding change of the initial phase. So, for instance, if harmonic oscillation is described by an expression (1) = A sin(cot + (p ), it is also possible to present it in the form (1) = A co (cot+cpi + n/2) = A co (col+(p, where P2 = Pi + nil. 

Let us make an interconnection between the angular frequency to, the frequency v and the period T. Sine and cosine are periodical functions with period In. This means that after the time interval T a system returns to its initial state and the phase is changed to In, i.e., [co(t + T)q) — u)t + (p] = In. Thereby, the angular frequency is connected with period T and frequency v by the expressions 

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