## Thursday, March 3, 2011

### Ocillations

Oscillations occur when a system is disturbed from a position of stable equilibrium. This displacement from equilibrium changes periodically over time. Thus, Oscillations are said to be periodic, and display periodic motion. Oscillations are very common in everyday life with familiar examples being the motion of a clock pendulum or the vibrations of strings on musical instruments. Oscillations are also important in many mechanical systems in the real world such as a car suspension. It is thus very important to be able to study and understand these mechanical systems in order to control them in critical situations.

Notice that for a system to be oscillating, the shape of the displacement - time graph does not matter. The only Property that matters is that the motion is periodic.

Basic properties of Oscillating Systems

It is important for us to understand the basic properties of oscillating systems if we are going to be able to understand the system as a whole. The first of these properties we must understand is the Amplitude of the oscillation. The amplitude of the oscillation is the parameter that varies with time and this resides on the y-axis of the oscillation graphs. In figure 1, the amplitude of the oscillation is the displacement of the object from its equilibrium position - however this is not always the case. In other systems, such as electric fields, the amplitude of the oscillation is the intensity of the electric field as it is the intensity that varies with time.

Another important property of an oscillation system is the Time Period (T) of the oscillation. The time period of the oscillation is simply the time taken for the oscillation to repeat itself. That is, it is the time between successive oscillations of the system (see figure 2). The other basic property of an oscillating system is the frequency, which is closely related to the time period. As we know, one complete oscillation of the system is defined by the time period, T and is known as 1 cycle. The frequency of the oscillating system is simply the amount of cycles that happen in 1 second. So,
Formula

The units of frequency are cycles per second which are given the name Hertz (Hz).

Further Properties of Oscillating Systems

Angular Frequency

The Angular Frequency of a system is the rotational analogue to frequency. It is given the symbol ω and is measured in radians per second (rads-1). It is defined by the equation
Formula ω = 2#/T

but, f = 1/T

and so is related to frequency by ω = 2#f

Phase

The Phase of an oscillation is the amount the oscillation lags behind, or leads in front of a reference oscillation. For example, take a sine oscillation of maximum amplitude, A, and angular frequency, ω, and also a cosine oscillation of maximum amplitude, A, and angular frequency, ω .

Now, we can take the sine wave to be our reference oscillation. It can be seen from the diagram that the cosine wave lags behind the sine wave by π/2 (1/4 of a wavelength). So, we can say that the two waves are out of phase by π/2 or that there is a phase difference of π/2. Oscillations can have phase differences of any multiple of π. However, if they have a phase difference of either 0 or 2π they are said to be in phase.

Harmonic Oscillations

Harmonic oscillations are just oscillations that are made up of sine and cosine varying waves. They make up the largest group of oscillations that exist. Any oscillation that varies with a sine or cosine function, or both, is said to be a harmonic oscillator. For a harmonic cosine oscillator, with maximum amplitude, A, and angular frequency, ω, the amplitude at anytime is given by y= A cos ωt. Similarly, for a harmonic sine oscillator, with maximum amplitude, A, and angular frequency, ω, the amplitude at anytime is given by y= A sin ωt.