How Does the Average Frequency (w1+w2)/2 Affect Undamped Forced Vibrations?

[aritra]

as we know ,the equation to the undamped forced vibration is
x=Asin{(w1+w2)/2}sin{(w1-w2)/2},the symbols have their usual meaning..but i can't understand how (w1+w2)/2 comes in picture in the phasor diagram...also its significance

 

[aritra]

please reply

 

[ravenprp]

The equation for undamped forced vibration, x=Asin{(w1+w2)/2}sin{(w1-w2)/2}, represents the displacement of a vibrating object at any given time. The symbols in the equation have their usual meanings, where A is the amplitude, w1 is the natural frequency of the system, and w2 is the frequency of the external force.

The term (w1+w2)/2 in the equation represents the average frequency of the system. This average frequency is important because it determines the overall behavior of the system. When the average frequency is close to the natural frequency of the system, the amplitude of vibration will be large, resulting in a phenomenon known as resonance. This can be seen in the phasor diagram, where the amplitude of the vibration is represented by the length of the phasor.

On the other hand, if the average frequency is significantly different from the natural frequency, the amplitude of vibration will be smaller. This is because the external force is not in sync with the natural frequency and is unable to transfer energy efficiently to the system.

In summary, the term (w1+w2)/2 in the equation and its representation in the phasor diagram help us understand the behavior of the system and how the external force affects it. It is a crucial factor in determining the amplitude of vibration and can lead to resonance or dampening of the vibration.