User's Manual

38 The Editor View
air which will propagate to our ears as sound waves.
Mathematically, a complex vibrational motion can be decomposed into elementary motion pat-
terns called the normal modes of the object. Under a normal mode, all the parts of the structure
move in phase and at the same frequency in a sinusoidal motion. In other words, this complex
motion results from the fact that objects naturally oscillate at many different frequencies at once,
each frequency being related to a normal mode of vibration. These frequencies are called partials;
the lowest partial is called the fundamental and the higher ones are referred to as overtones. When
relating to music, the fundamental corresponds to the note played and the overtones are called
harmonics as in most musical instruments their frequency is a multiple integer (or almost) of the
fundamental.
As an example, the vibration motion associated with two normal modes of a rectangular plate is
illustrated in Figures 21 and 22. In the first figure, one can see the vibration motion associated with
two different normal modes of the plate (modes [1,1] and [3,2]). Over one period of oscillation,
all the points go up and down in phase. The principle remains the same for all mode, the motion
pattern only becoming more and more complex as the order of the mode increases. The full motion
of a plate, however complicated, will always correspond to a combination of all its normal modes.
Figure 22 is a top view of the plate and shows contour lines corresponding to the same normal
modes. A contour line groups points that oscillate with the same amplitude. In particular, the
straight lines in the second graph of this figure, corresponds to so-called nodal lines where the
amplitude of the motion is zero and therefore where the plate is still.
The relative frequencies or ratio of the frequency of the overtones to the fundamental frequency
is specific to the type of the object and its boundary conditions (whether its boundaries are free to
vibrate or are fixed). In other words this distribution of partials is characteristic of the type of object
and could be viewed as its tonal signature; it allows us to distinguish, for example, a vibrating
plate from a drumhead. The specific frequency of the partials, related to the sensation of pitch, is
determined by the dimensions of the object, for example a small plate will have a higher pitch than
an larger one.
Figure 21: Motion corresponding to normal mode [1,1] and [3,2] of a plate.