**Hardening and
softening nonlinear behavior in acoustic resonators**

By Bart Lipkens [ from ASA’s Echoes from Austin ]

We have found that it is possible to create very large amplitude shock-free standing waves in shaped acoustic resonators. Overpressures of 400 of ambient pressure have been measured. An interesting nonlinear behavior is observed when measuring the frequency response functions of these shaped cavities. First we introduce some of the terminology related to nonlinear oscillations by referring to the mechanical vibrations of a simple mass-spring system. Then we take a closer look at the nonlinear behavior of the shaped acoustic resonators.

Nonlinear or amplitude dependent, behavior of a mechanical oscillator such as a simple mass-spring system can be demonstrated by considering an amplitude dependent spring stiffness. A hardening spring behavior is observed when spring stiffness increases with displacement amplitude, whereas softening corresponds to a spring stiffness that decreases with amplitude. The Duffing equation is the standard equation that describes this nonlinear spring behavior.

The frequency response function of a hardening spring is amplitude dependent. With increasing amplitude the resonance frequency increases. The response curve becomes asymmetric and "leans" towards higher frequencies. Eventually, the curve becomes multi-valued and hysteresis occurs. A similar curve is obtained for the softening spring, but this time "leaning" is towards lower frequency values. Now let's look at the nonlinear behavior of acoustic standing waves in cavities. First let's focus on a cylindrical cavity. It is well known that at elevated amplitudes the propagation is nonlinear. An initially sinusoidal waveform distorts at elevated amplitudes and eventually becomes a sawtooth waveform, i.e., a shock followed by a smooth expansion. Once shock formation occurs, the peak pressure of the sawtooth saturates, i.e., reaches a maximum value. Any additional energy supplied to the wave is dissipated as heat across the shock front. Saturation occurs in a cylinder at peak levels of about 10 of ambient. Insight into this behavior is gained from inspection of the modal spectrum of the cylinder. For a cylinder the modal spectrum is consonant, i.e., the higher order modes are all integer multiples of the fundamental resonance frequency. For the cylinder the harmonic frequencies generated by nonlinearity for an excitation at the fundamental resonance frequency are coincident with the modal frequencies. This enables efficient transfer of energy from the fundamental frequency to the higher harmonics, and ultimately leads to shock formation. The frequency response curves for a cylinder are symmetric and do not show any hardening or softening behavior. Now let's take a look at shaped resonators. For shaped resonators the modal spectrum is dissonant, i.e., the higher order modal frequencies are no longer multiples of the fundamental frequency. Two resonators were used in this study, the first is a cone, and the second is a bulb shape. For the measurements the resonators were filled with refrigerant R- 134a. Since the modal spectrum is dissonant, it follows that the harmonic frequencies are not coincident with the modal frequencies of the cavity. The growth of the harmonic amplitudes is therefore less than that for the cylinder. The frequency response function of the conical resonator is shown in Fig. 1.

The response is that of a hardening nonlinearity, i.e., resonance frequency increases with amplitude and the curve "leans" towards higher frequencies. At high amplitudes the curve becomes multi-valued and hysteresis effects occur.

The frequency response function for the bulb resonator is shown in Fig. 2. A softening behavior is observed for the bulb.

The interesting fact is that whereas for the mechanical system we had to change the material property to change the nonlinear behavior, it is sufficient to change the boundary of the resonator to alter the nonlinear behavior of the cavity. An explanation of the behavior is gained from an inspection of the modal spectrum. For the cone resonator the growth of the second harmonic is dominated by the contribution of the second mode This typically creates a u-shaped waveform with narrow peaks and broad valleys. This wave shape exhibits hardening behavior. On the contrary, for the bulb resonator the second harmonic is dominated by the third mode and this creates an m-shaped waveform, i.e., broad peaks and sharp minima. Usually, softening behavior is observed.

In conclusion, frequency response curves of acoustic resonators are discussed. Both hardening and softening nonlinear behavior of acoustic resonators is reported. Changing resonator shape can change the nonlinear behavior as shown by the frequency response curves.

References

1.Yu. II'inskii, B. Lipkens, and E. Zabolotskaya, "Energy losses in an acoustical resonator," J Acoust. Soc. Am. 109, 1859-1870 (2001).

2.Yu. II'inskii, B. Lipkens, T. S. Lucas, T. W Van Doren, and E. Zabolotskaya, "Nonlinear standing waves in an acoustical resonator'', J. Acoust. Soc. Am. 104, 2664-2674 (1998).

3.C. Lawrenson, B. Lipkens, T. Lucas, D. Perkins, and T. Van Dorm, "Measurements of macrosonic standing waves in oscillating cavities;' J Acoust. Soc. Am.,104, 623-636 (1998).

4, M. Hamilton, Yu. Ilinskii, and E. Zsbolotskayn, "Linear and nonlinear frequency shifts in acoustical resonators with varying cross-sections," J Acoust. Soc. Am., 110, 109-119 (2001)

Bart Lipkens is an Assistant Professor of Mechanical Engineering at Western New England College in Springfield Massachusetts. He is a member of the Physical Acoustics Technical Committee. This article is based on his presentation at the Austin ASA meeting.