# Theory of Forced Oscillations

We shall show here the relevant theoretical aspects related to forced oscillations. Our working example will be the mass-spring system. In general we are dealing with an oscillatory system, with losses and forced to vibrate by an external action that shall be supposed to be harmonic.

The losses are proportional to the velocity through a coefficient equal to 2bm. To link the friction coefficient with the losses that can be implemented in xyZET, the following relation applies:

### Amplitude and phase versus frequency

The equation corresponding to forced oscillations, as we have previously seen, is:

Where w 0 is the natural frequency of the system and b represents the friction. This equation has a general solution composed of two independent terms.

• A transient that damps out in a time in the order of some 1/(2b) time units. This transient strongly depends on the initial conditions of the system.
• A stationary (steady state) part, of the form:

An example of this motion is shown in experiment

where the transient and steady state are clearly identifiable.

This last, the steady state, is the part on which we shall concentrate our attention. One thing that has to be stressed is that the response of the oscillator is at a frequency corresponding to the one of the external agent. This is true and characteristic for all linear systems. The natural frequency of the system is involved in the amplitude of the oscillation, A, and in the phase delay, d, between the elongation x and the external force.

The values for A and d for the steady state can be deduced after simple substitution of x=Asin(wt-d) in the differential equation. After some manipulations we get:

for the amplitude. This equation tells us that the largest amplitude is achieved when the system is set to resonance, that is to say, when w = w 0 . If the losses (b) are small, the amplitude can be very large at resonance and the system could eventually be destroyed; in this case the system hardly responds to the external force when the external frequency deviates from the frequency of the system. That is to say, the changes around the resonance are abrupt when the losses are small and, conversely, very slow when the system is a very damped one.

For the phase we get:

If the damping is small, the phase shift, d, between the external force and x is:

• Close to 0º when w < w 0 .
• Close to 180º for w > w 0 .
• 90º for w = w 0 .
This change is sharp when the losses are small. The transient from 0 to 180º is more gradual when the losses are large.