In practice, it often causes fluctuations against each other. For example, If on the floor there are two motor, working at different speeds, then Paul commits complex oscillation, which results from the blending of the oscillations, caused by the operation of each engine separately. The resulting wobble can be quite challenging on your mind.

Consider the addition of harmonic oscillations of the same frequency, that occur on the same line. Such fluctuations can be added graphically. Having graphics offset x’ and x” Depending on the time t, timetable result offset fluctuations can be found algebraically adding offsets x’ and x” in every moment of time (in figure. 24.19, and shows this disjunction on two points). Thus, the resulting offset x at each point is determined by the ratio of:

x = x ' + x”.

Connecting the ends of the curve obtained by ordinate x, find a schedule of the resulting oscillations.

As can be seen from figure. 24.19, and, If you add the same harmonic frequency harmonic oscillation is produced the same frequency. The addition of such fluctuations can be performed easier, without resorting to graphs.

Let folding fluctuations are described by equations:

x’ = A'sin(ϕ'₀+2πvt), x ' = a ' sin(ϕ''₀+2πvt)

In the theory of oscillations is proved, that amplitude and initial phase of the resulting fluctuations in x = Asin(ϕ₀+2πvt) You can find, folding vector amplitude and’ and while”:

A = A’ + And”.

This is done as follows:. From an arbitrary point O (rice. 24.19, b) carry out horizontal poluprâmuû, which are the initial phase. From point O spend vectors and’ and (A)”, the provisions which define the initial phase ϕ '₀ and ϕ '₀. The amplitude of the resultant vibrations and is the diagonal of the parallelogram, built on the vectors and’ and (A)”, and the angle ϕ₀ Specifies the starting phase of the resulting oscillations. In figure. 24.19, b shows the vector diagram hesitation, graphs which depicted on Figure. 24.19, and.

Note, What A ’, A” and (A) is a movable radii folded variations and the resulting fluctuations, and their vectors are called conditionally, because they are not relevant to the concept of vectors as physical value. In figure. 24.19, bthey are depicted in the initial time. (Show, что при вращении этих векторов с угловой скоростью ω (counterclockwise) their projection on a vertical direct define appropriate offsets.)