Measurements of nonlinear effects in a driven vibrating wire

Abstract

Experimental measurements have been made of the motion of a red brass harpsichord wire driven electromagnetically in a fixed direction perpendicular to the equilibrium position of the wire. The motion is complex compared to that predicted by simple linear theory because of effects due to tensional changes and longitudinal motions. Optoelectronic detectors are used to measure amplitude and phase of the transverse motions as functions of the driving frequency, both in the driving direction y and the direction z perpendicular to y. Near the free-vibration fundamental frequency f0 the z and y amplitudes are comparable even for a very low driving force and amplitude. Amplitude jumps and hysteresis effects are observed for large amplitudes. The z–y phase difference is measured as 0°, 90°, and 180° in different frequency regions, yielding both planar and whirling or tubular motion. As the driving frequency increases, the phase difference between the driving force and the y motion varies steadily from 0° to 90° before jumping to 180°. There is no evidence of a critical frequency of onset of the z motion as is predicted in some theoretical treatments. Similar effects are observed near 3f0, for which amplitude measurements have been made down to 0.01 μm for a 0.71-m-long wire.

Nonlinear forced vibrations of thin structures with tuned eigenfrequencies: the cases of 1:2:4 and 1:2:2 internal resonances

“…Secondly, the particular sound of some musical instruments can be explained by some nonlinearities, such that they appear as a case where nonlinear resonances between numerous eigenfrequencies should be key to properly understand their dynamical behavior. The string is obviously the most common case for string instruments sharing the two properties of nonlinear vibrations together with commensurable eigenfrequencies; see, e.g., [12,17,18,32]. Nonlinearities are also encountered in reed instruments such as saxophone and clarinette; see, e.g., [27] and brass instruments (see, e.g., [20]) and references therein.…”

Parametric resonance in non-linear elastodynamics

“…For the neo-Hookean material (β = 1/2) we obtain the Hill equation 9) and the related Mathieu equation .7) is unstable for any value of * . In figure 1 we plot, for different values of λ S , v R versus β at the 2 : 1 resonance.…”

“…In such an experiment a periodic change in the longitudinal tension of a taut string resonantly parametrically excites transverse waves when the frequency of the change of the tension is close to twice the natural frequency of the transverse waves. The simplicity of this physical system has generated an enormous literature from the experimental (e.g., Hanson et al, 1994) and theoretical side (e.g., Rowland, 1994), starting from the considerations of Lord Rayleigh summarized in his book on acoustics (Strutt, 1945).…”

Modeling of tension modulation nonlinearity in plucked strings

“…Legge and Fletcher described the coupling of vibrating modes in a one-polarization wave motion and demonstrated the generation of missing harmonics as a result of this coupling [19]. Hansen et al experimented with coupling of polarizations and reported measured results of amplitudes and phase differences of transversal components under forced motion [20]. More interestingly, they found nonlinear coupling of the two transversal polarizations even at vibration displacements of only a few microns.…”

“…For the single-delay-loop model, the varying delay is . By substituting for and for in (17), we obtain (20) Parameter may be solved from (20) as (21) Similarly, we may estimate the maximum average elongation using (21) if is known. An example of the determination of the parameter follows.…”