Periodic self-modulation of an electrodynamically driven heated wire near resonance

Abstract

A thin nichrome wire driven near resonance by the Lorentz force and heated by an alternating electrical current is a popular lecture demonstration. Due to the convective cooling of the portions of the wire moving with the greatest amplitude, only glowing regions near a velocity node will be visible in a darkened room. Nonlinear effects and the thermal expansion coefficient of the wire displace the wire's tensioning mass. By adiabatic invariance, the work done on or by the vibrating wire, due to the changes in the mass's elevation, causes the natural frequency of the standing wave resonance to be shifted. Competition between the thermal inertia of the wire and the convective heat transfer coefficient introduces an exponential thermal relaxation time so that the amplitude of vibration is dependent on the ratio of the drive frequency to the changing resonance frequency at an earlier (retarded) time. These thermal and kinetic effects are incorporated into three coupled nonlinear ordinary differential equations that are separated by the method of multiple time scales and are solved numerically, reproducing both the spontaneous appearance of stable periodic amplitude modulation and the hysteretic behavior observed with increasing or decreasing of the drive frequency.

String Theory

“…(3.51). That can be done analytically 11 or by use of an equation solver. I chose the latter and found β ¼ 0.6385, making ω o 2 ¼ 1.4460 (g/L ).…”

“…Of course, a real string, driven by a real displacement drive, will not produce infinite displacements. Long before the displacements become infinite, we will have exceeded the limitations of our linear approximation, and the average tension in the string will start to increase with increasing amplitude, causing the speed of the transverse vibrations to increase, thus de-tuning the resonance condition [11]. Also, practical considerations, including the effective output mechanical impedance of the driver and dissipation in the string, radiation of sound from the string, and losses introduced by motion at the "fixed" end, will all limit the transverse displacement amplitudes.…”

“…, and ℓ ! are all mutually perpendicular [11]. If the electrically conducting string is fixed at both ends and driven by the current, I(t), applied at a frequency that would excite one of the fixed-fixed string's normal modes (i.e., f n ¼ nc/2 L ), then if ℓ !…”

“…When the fraction is written as the ratio of two quadratic functions in the form u¼ (A þ 2Hm þ Bm 2 )/(a þ 2hm þ bm 2 ),then the minimum or maximum value is given by the solution of the following quadratic equation:(abh 2 )u 2 -(aB þ bA À 2hH)u þ (AB -H 2 ) ¼ 0 11. If you are too lazy to work out the derivative (or want an independent check), it is possible to use an equation solver, like that available in many mathematical software packages, to determine the value of m which minimizes Eq.…”