Exploring the resonant vibration of thin plates: Reconstruction of Chladni patterns and determination of resonant wave numbers

Exploring the Origin of Maximum Entropy States Relevant to Resonant Modes in Modern Chladni Plates

Abstract: The resonant modes generated from the modern Chladni experiment are systematically confirmed to intimately correspond to the maximum entropy states obtained from the inhomogeneous Helmholtz equation for the square and equilateral triangle plates [...]

Deviations from Poisson statistics in the spectra of free rectangular thin plates

The counterintuitive fact that wave chaos appears in the bending spectrum of free rectangular thin plates is presented. After extensive numerical simulations, varying the ratio between the length of its sides, it is shown that (i) frequency levels belonging to different symmetry classes cross each other and (ii) for levels within the same symmetry sector, only avoided crossings appear. The consequence of anticrossings is studied by calculating the distribution of the ratio of consecutive level spacings for each symmetry class. The resulting ratio distributions disagree with the expected Poissonian result. They are then compared with some well-known transition distributions between Poisson and the Gaussian orthogonal random matrix ensemble. It is found that the distribution of the ratio of consecutive level spacings agrees with the prediction of the Rosenzweig-Porter model. Also, the normal-mode vibration amplitudes are found experimentally on aluminum plates, before and after an avoided crossing for symmetrical-symmetrical, symmetrical-antisymmetrical, and antisymmetrical-symmetrical classes. The measured modes show an excellent agreement with our numerical predictions. The expected Poissonian distribution is recovered for the simply supported rectangular plate.

High-power structured laser modes: manifestation of quantum Green's function

The distributions of resonant frequencies in an astigmatic cavity are theoretically confirmed to be analogously equivalent to the quantum energy structures of two-dimensional commensurate harmonic oscillators. In the first part [Opt. Lett.45, 4096 (2020)OPLEDP0146-959210.1364/OL.399251] of this two-part series study, the lasing modes were verified to reveal a variety of vortex array structures. Here, in the second part of this two-part series study, the lasing modes are confirmed to agree very well with the quantum Green's functions that correspond to a bundle of Lissajous figures in the high-order regime.

Motion of Heavy Particles on a Submerged Chladni Plate

Heavy particles are traditionally believed to gather at the nodes of a resonating plate, forming standard Chladni patterns. Here, for the first time, we experimentally show that heavy particles, i.e., sub-mm particles, can move towards the antinodes of a resonating plate. By submerging the resonating plate inside a fluidic medium, the acoustic radiation force and the lateral effective weight become dominant at the sub-mm scale. Those forces, averaged over a vibration cycle, move the particles towards the antinodes and generate sophisticated patterns. We create a statistical model that relates the complex motion of particles to their locations and plate vibration frequencies in a wide spectrum of both resonant and nonresonant frequencies. Additionally, we employ our model to control the motion of single particles and a swarm of particles on the submerged plate. Our device can move particles with sufficient power at an exceptionally wide frequency range, potentially opening a path to new particle manipulation techniques at sub-mm scale in fluidic media.

Point-driven modern Chladni figures with symmetry breaking

Point-driven modern Chladni figures subject to the symmetry breaking are systematically unveiled by developing a theoretical model and making experimental confirmation in the orthotropic brass. The plates with square shape are employed in the exploration based on the property that the orientation-dependent elastic anisotropy can be controlled by cutting the sides with a rotation angle with respect to the characteristic axes of the brass. Experimental results reveal that the orientation symmetry breaking not only causes the redistribution of resonant frequencies but also induces more resonant modes. More intriguingly, the driving position in some of new resonant modes can turn into the nodal point, whereas this position is always the anti-node in the isotropic case. The theoretical model is analytically developed by including a dimensionless parameter to consider the orientation symmetry-breaking effect in a generalized way. It is numerically verified that all experimental resonant frequencies and Chladni patterns can be well reconstructed with the developed model. The good agreement between theoretical calculations and experimental observations confirms the feasibility of using the developed model to analyze the modern Chladni experiment with orientation symmetry breaking. The developed model is believed to offer a powerful tool to build important database of plate resonant modes for the applications of controlling collective motions of micro objects.

Controlling the motion of multiple objects on a Chladni plate

The origin of the idea of moving objects by acoustic vibration can be traced back to 1787, when Ernst Chladni reported the first detailed studies on the aggregation of sand onto nodal lines of a vibrating plate. Since then and to this date, the prevailing view has been that the particle motion out of nodal lines is random, implying uncontrollability. But how random really is the out-of-nodal-lines motion on a Chladni plate? Here we show that the motion is sufficiently regular to be statistically modelled, predicted and controlled. By playing carefully selected musical notes, we can control the position of multiple objects simultaneously and independently using a single acoustic actuator. Our method allows independent trajectory following, pattern transformation and sorting of multiple miniature objects in a wide range of materials, including electronic components, water droplets loaded on solid carriers, plant seeds, candy balls and metal parts.

Manifesting the evolution of eigenstates from quantum billiards to singular billiards in the strongly coupled limit with a truncated basis by using RLC networks

The coupling interaction between the driving source and the RLC network is explored and characterized as the effective impedance. The mathematical form of the derived effective impedance is verified to be identical to the meromorphic function of the singular billiards with a truncated basis. By using the derived impedance function, the resonant modes of the RLC network can be divided into the open-circuit and short-circuit states to manifest the evolution of eigenvalues and eigenstates from closed quantum billiards to the singular billiards with a truncated basis in the strongly coupled limit. The substantial differences of the wave patterns between the uncoupled and strongly coupled eigenmodes in the two-dimensional wave systems can be clearly revealed with the RLC network. Finally, the short-circuit resonant states are exploited to confirm that the experimental Chladni nodal-line patterns in the vibrating plate are the resonant modes subject to the strong coupling between the oscillation system and the driving source.