The nonlinear vibration of plates is what leads to the characteristic sounds of gongs—pitch glides, crashes and swells—that we all know. A linear model is insufficient to capture these effects, which vary strongly with the striking force. This particular nonlinearity is a rather complex one—the simplest suitable model is that of Föppl and von Kármán, and has been used for offline sound synthesis in the past [1]. One of the main features is the spontaneous generation of new frequency components—leading ultimately to noise in the limit of very hard strikes. See the videos below, illustrating the vibration of a plate, initialised in its lowest mode shape, at various amplitudes.

It’s also useful to see the effect of the nonlinearity through a spectrogram. See the series of images below, illustrating the emergence of pitch glides, as well as noise at increasing strike amplitudes, in Newtons, for a steel plate of thickness 0.5 mm.

Recently, advances in algorithm efficiency [2] have made it possible to achieve sound synthesis in real-time—the nonlinearity can be dealt with fully explicitly. This is done through so-called invariant energy quadratisation (IEQ) [3] and scalar auxiliary variable (SAV) approaches [4] that also allow conditions on numerical stability—a big concern for strongly nonlinear systems like the gong! Another ingredient is a fast solver for the biharmonic equation, developed by our collaborator Zehao Wang at the University of California San Diego [5], and based on earlier fast matrix inversion techniques [6].

Here, we’ve developed a full real-time model of a gong, including the nonlinearity, a striking force, and losses. We’ve also built in variable output locations—so you can emulate the phasing effects you hear in a real gong that is able to exhibit rigid body motion relative to a listener. This is done by “scanning” output locations over the plate at a subaudio rate.

Here are a few sound examples. All are for a steel plate, of thickness 0.5 mm. Output is stereo, and the rate at which the output locations “scan” over the late surface is 0.2 Hz. The sample rate is 44.1 kHz.

We’re working on turning this into a playable instrument. Stay tuned!

References:

[1] S. Bilbao, “Sound synthesis for nonlinear plates,” in Proc. 8th Int. Conf. Digital Audio Effects, Madrid, Spain, Sept. 2005, pp. 243–248

[2] S. Bilbao, M. Ducceschi, and F. Zama, “Explicit exactly energy-conserving methods for Hamiltonian systems,” J. Comp. Phys., vol. 427, pp. 111697, 2023

[3] J. Zhao, “A revisit of the energy quadratization method with a relaxation technique,” Appl. Math. Lett., vol. 120, pp. 107331, 2021

[4] M. Jiang, Z. Zhang, and J. Zhao, “Improving the accuracy and consistency of the scalar auxiliary variable (SAV) method with relaxation,” J. Comp. Phys., vol. 456, pp. 110954, 2022

[5] Z. Wang and M. Puckette, paper in preparation, Stockholm Musical Acoustics Conference, Stockholm, Sweden, 2023

[6] B. Buzbee, G. Golub, and C. Nielson, “On direct methods for solving Poisson’s equations,” SIAM J. Num. Anal., vol. 7, no. 4, pp. 627–656, 1970