Helmholtz resonators

Have you ever blown across the top of a bottle to make a sound? If you have, you probably do not know that this intriguing phenomenon is called Helmholtz resonance, and it is due to the resonance of a fluid in a cavity. If you did know it, then you may find the following article quite interesting.

Helmholtz resonators are acoustic passive devices that leverage this resonance phenomenon. They are widely used as a convenient and effective way of reducing noise in a narrow band of frequencies. As such, they have been considered for use in fluid power systems for reducing the pressure ripple generated by positive displacement pumps [1]. They are branched away from the main pipeline and they are typically applied to damp out the pressure ripples caused by the hydraulic pumps. Provided that the dimensions of the Helmholtz resonator are smaller than the acoustic wavelength, its dynamics behavior can be described as a lumped system, namely a spring-mass vibration absorber, as depicted in the picture below.

Developing further this analogy, it is possible to express the device resonance frequency as shown in the equation below [2]:

Knowing the target frequency of the noise to suppress, it is possible to select a combination of geometrical parameters that satisfy the equation above. For example, let us suppose we want to filter a frequency of 118 Hz in a hydraulic system pressurized at 3000 psi and filled with a hydraulic oil such as Skydrol (the speed of sound in these conditions is 1180 m/s). Then, a possible combination of values that allows to suppress such frequency is:

I used the simulation software tool Simcenter Amesim [3] to create a model of a simple straight hydraulic line with the following properties:

Simcenter Amesim models pipes using a lumped approach to account for fluid and pipe or hose capacity (including wall compliance), inertia and friction. Also, it embeds frequency dependent friction and, for this reasons, it is particularly suitable to study phenomena related to wave dynamics. Using the tool's linear analysis capabilities, I plotted the system response in a Bode plot in two configurations: with and without the Helmholtz resonator in red and blue respectively.

It is striking to see how the resonance frequency at 118 Hz was effectively suppressed, passing from a peak of 21dB to -52dB. It is also worth noting that the introduction of the resonator modified the entire system's frequency response, displacing lower and higher order natural frequencies.

This simulation model allows to overcome a limitation that is embedded in the derivation of the theoretical formula described above. In fact, the equation does not take into account friction effects that take place in the hydraulic lines, and this can result into an offset with respect to the real resonance frequency. This offset is proportional to the resonator neck's aspect ratio (L/D). Let us consider another resonator, with a different set of geometrical value, that differs from the one presented above only in that it has a higher neck aspect ratio, but with the same theoretical frequency, as summarized in the table below:

If we compare the simulation results obtained for the two resonator, we obtain the following Bode plot (red line: original resonator, blue line: resonator with higher aspect ratio). We can clearly see that, because of the non negligible friction effects taking place in the resonator's neck, the frequency response of the system is now different.

I hope this article gave you some interesting insights on what are Helmholtz resonators, how they can be used to suppress fluid born noise, and how system simulation can help overcome limitations found in common theoretical approaches.

References:

[1] Nicholas E. Earnhart & Kenneth A. Cunefare (2012) Compact Helmholtz Resonators for Hydraulic Systems, International Journal of Fluid Power, 13:1, 41-50, DOI: 10.1080/14399776.2012.10781045

[2] T. J. Viersma. Analysis, Synthesis, and Design of Hydraulic Servosystems and Pipelines. Elsevier Scientific Publishing Company, 1980.

[3] https://www.plm.automation.siemens.com/global/en/products/simcenter/simcenter-amesim.html