FORCED OSCILLATOR

Every physical system has one or more natural vibration frequencies and one or more natural modes of energy dissipation (damping) that characterize its behavior when stimulated by a transient impulse.  These damped oscillations will eventually cease unless energy is supplied to sustain them.  If driven by a series of external impulses at some arbitrary frequency, the system will necessarily vibrate at the driver frequency, but with an amplitude and phase that depend not only on the driver amplitude and frequency, but also on the natural frequencies and damping constants of the system.  If the frequency of the external stimulus is varied, the amplitude of the response at or near one of the natural frequencies can exceed the amplitude of the driver, a condition known as resonance.  The amount of damping controls the size of the resonant response, the sharpness of the resonance as a function of frequency, and the phase shift between the stimulus and the response.  For small damping the amplitude is maximum when the frequencies of the driver and the oscillator coincide, but for large damping there is a frequency shift.  If the damping time of the oscillator is large, the resonance width is small, and vice versa.  Thus the product of the frequency width and the damping time is of order unity.  This property is known as the uncertainty principle, and expresses the impossibility of measurements that are arbitrarily precise in both time and frequency.

If the driver and response frequencies are equal but 180 degrees out of phase, the stimulus tends to suppress the response oscillations (e.g. the driven tuned mass-dampers in tall buildings that are used to decrease their sway during high winds).  If the driver exactly compensates for the energy dissipated by damping, it will sustain the motion (e.g., the escapement in a pendulum clock).  If the driver supplies more energy than is dissipated by damping, the amplitude of the system will increase dramatically, and perhaps catastrophically.  Many examples of resonance exist, such as the shattering of a wine glass by sound, the use of a microwave oven to drive electrons in water molecules at their natural frequencies, the use of a tuned electrical circuit to filter electromagnetic waves detected in a radio receiver, and the bridge collapses that have occurred when marching troops failed to break cadence.  In some cases the external driving force can by initiated by the presence of the responding oscillator itself.  For example, alternating vortex swirls can be formed when a fluid flows past an object, which can exert a periodic force on the object that forms them.  When the resonance condition is met, huge amplitudes have be observed (e.g., the collapse of the Tacoma Narrows bridge in 1940 and the failure of aircraft wings in the 1960's).

Another application of driven oscillation is given by resonance fluorescence, which is used as a tool in the study of atomic structure.  In one implementation of this technique, observation of the absorption and reemission of laser light of known but variable frequency by an atomic sample permits the determination of the natural frequencies (that characterize the energy level separations) and damping constants (that characterize the level lifetimes) of the atoms.

Lorenzo J. Curtis

Department of Physics and Astronomy

University of Toledo

Topic words, in order of first occurrence:

Natural Frequency; Vibration; Damping; Impulse; Amplitude; Phase; Resonance; Uncertainty Principle; Mass; Pendulum; Molecule; Vortices; Fluorescence; Laser; Atom.