Mechanical Vibrations and their Characteristic Modes
Sometimes the reasons for the vibration are far from obvious. Furthermore the motion may be irregular and bear little resemblance to the motion of small amplitude vibrations induced by an elastic spring. In such cases no matter how long one examines the vibration no definite frequency or amplitude emerges from the complex motion.
Before discussing this question let us explore the ability of systems to vibrate freely after they have been disturbed in some way. The initial disturbance can be regarded as a means of transferring energy to the system from some external agency. The components of mechanical systems possess mass which acquire kinetic energy when in motion. The forces on them arise from both internal and external sources. Examples of external forces include effects of gravitation and the presence of a fluid environment such as the air or sea. Internal forces arise from the effects of elastic forces such as those in evidence in the deformed spring. The work done in making small deformations in a solid elastic body can sometimes be recovered when the deformation relaxes. Such solids therefore have the ability to store energy in an elastic deformation. Sometimes the system changes its shape without changing its volume. Energy is then expended by internal elastic stresses to maintain the volume of the system constant. In general the total energy becomes shared between the energy of motion and the work required to maintain the configuration of the system as it evolves in time. In all mechanical systems there is a loss of energy that is dissipated as heat energy into the environment. The energy fed into a system is distributed between the kinetic, elastic and dissipative channels at each instant of time. Given an arbitrary initial disturbance the ensuing motion is often quite complicated even for simple mechanical systems. However, if one excites the motion carefully, it may be possible to initiate a characteristic motion that does have a particular frequency of oscillation. Such a motion is associated with a particular geometric configuration or mode that characterises the motion. Even more surprising is the fact that (in principle) every system can be made to exhibit a particular characteristic mode from a family of different modes. Each mode has a definite characteristic frequency and the set of all such frequencies is called the normal mode spectrum. It constitutes a kind of "genetic blueprint'' associated with any system capable of vibration.
The normal mode spectrum depends only on the geometry and constitution (density, stiffness...) of the system. The geometry includes a specification of its extension in space and the manner in which it is attached at its extremities. Thus the spectrum of a vibrating drum head with a circular boundary will be different to one with an elliptical boundary. Furthermore if one presses ones finger at any point on the drum surface the boundary conditions are changed and the characteristic frequencies will change accordingly.
There is a rather special kind of system whose vibrational state can always be resolved into a superposition of component oscillations each of which corresponds to one of the individual normal modes. The precise amount of the individual modal content at any moment depends on how the system is excited. These situations often arise when one considers perturbations of a more complex system and offer the engineer a way to probe the normal mode structure experimentally.
An example of such a system occurs when a piano string executes small lateral vibrations under tension. The normal mode spectrum of a piano string is determined by its length, density and tension and the fact that it is fixed to a rigid frame at each end. For such a string the spectrum consists of equally spaced frequencies.
If the string is tuned to note A above middle C the spectrum begins 440 Hz, 880 Hz, 1320 Hz ... and when struck with a piano hammer its vibration can be resolved into a mixture of these pure tones. However the frequencies above 440 Hz rapidly die away since damping attenuates the overtones. Although the lowest mode dominates it is the presence of these overtones and the way they are initially excited that gives the piano its recognisable sound and distinguishes it say from a similar note plucked on a harp or bowed on a violin.
The
motion of the piano string displaces the surrounding air which then transmits a
pressure wave in all directions. When this gas disturbance encounters the small
bones in the inner ear of a human another vibration stimulates electrical
signals that flow to the brain and create the perception of sound. It is quite
remarkable that the normal mode signature of the vibrating source (the piano
string) can be maintained during this transformation to the human brain.
Because of the intimate correlation between the spectral signature of a
mechanical vibration and the geometry of the vibrating source it is in
principle possible to "hear the shape of a vibrating drum".In general as one increases the mass of the moving parts of a vibrating system the frequency of the normal mode oscillation tends to decrease. On the other hand increasing the "stiffness" of the system causes the same frequencies to increase. These simple observations have powerful implications for the control of unwanted vibrations.
The normal modes of an elastic solid give rise to characteristic motions in space. For a typical piano string the most dominant motion refers to a displacement of the string in a plane containing the length of the string. For a stiff elastic rod the motion can include not only these "lateral" vibrations but also twisting or "torsional" vibrations and "axial" motions along the length of the rod.
A vibrating plate offers a means to visualise its normal modes. Sand
sprinkled on a thin metal plate clamped centrally to remain horizontal can be
excited into beautiful patterns by vibrating the plate at specific
frequencies.The characteristic vibrations give rise to lines of minimal motion (nodal regions) which separate domains containing regions of maximal vibration. The Chladni patterns arise because the sand collects in regions of minimal vibration. By vibrating the plate (either using a bow or, as in this case, an electric motor) at different frequencies patterns, unique to that frequency, will appear. All real systems experience damping of one kind or other. It may arise from frictional effects due to relative motion between the system and its surroundings or damping due to losses experienced by the medium itself in its motion. All forms of damping dissipate energy and each characteristic mode has a characteristic dissipation time during which it sheds a certain proportion of the energy in that mode. For a system that can be regarded as a superposition of normal modes the high frequency modes are usually damped out before the lower ones.