Waves can go through more systems beyond our guess
The recent technological advances in mechanical systems and signal processing demand better vibration control of flexible structures. At the moment, active vibration control design based on modal analysis (modal control) is popular and has been thoroughly established; unfortunately, difficulties are encountered in controlling modally dense structures, because modal frequencies and modal shapes are extremely sensitive to modeling errors. For certain flexible structure types consisting of simple members, such as vibrating strings, beams, and flexural waveguides, the dynamic response can be described using wave motion. In a general wave analysis process, one first derives the secondary constants from the system dynamics, then subsequently evaluates the steady-state response for harmonic excitation (harmonic analysis). This corresponds to evaluating the transfer functions on a point on the imaginary axis s=jω. A correct answer is provided if the secondary constants are defined as analytic in the open right-half plane (RHP); however, the result is invalid if singularities exist there.
To address this, Wakayama University researchers: Professor Kenji Nagase, Ayato Doshita, and Takuya Midoro, proposed to investigate wave analysis and control of mono-coupled periodic mechanical systems. They aimed to resolve the aforementioned issues but in so doing, they ended up revealing that, in most cases, secondary constants can be defined as analytic in the open RHP and can satisfy several required properties. The original research article is currently published in the research journal, Mechanical Systems, and Signal Processing.
In their approach, the research team used the wave analysis and control of general mono-coupled periodic systems. To begin with, the team reviewed the wave analysis process for the uniform case and identified several required properties of the secondary constants for the wave interpretation. In the wave analysis, the researchers employed the transfer matrix (or cascade matrix) method that has been used extensively in the field. The coefficient matrix eigenvalues correspond to the propagation constants, while the elements of a diagonalizing transformation matrix were determined from its eigenvectors, composing the characteristic impedances. Moreover, the team considered these secondary constants in the Laplace transform domain and defined them as analytic functions of s.
The authors demonstrated that the propagation constants were located inside and outside of the unit circle, and the characteristic impedances were positive real functions. These results justified the harmonic analysis in the wave analysis and guaranteed the closed-loop stability of the impedance matching controller. Moreover, the analyticity of the characteristic impedances was seen to guarantee the approximation error of the rational impedance matching controller, obtained using complex curve-fitting on the imaginary axis.
In summary, the study considered the wave analysis and control of mono-coupled periodic mechanical systems. The approach employed first considered the uniform case and precisely investigated the properties of the propagation constants and characteristic impedances as analytic functions. In a statement to Advances in Engineering, Professor Kenji Nagase mentioned that although they derived the uniformly varying condition heuristically in the non-uniform system analysis, other system types can also be analyzed using the wave analysis. Moreover, he added that experimental validation for practical systems is also important, and the proposed procedure can be applied to more general multi-coupled periodic systems.
Kenji Nagase, Ayato Doshita, Takuya Midoro. Analytical properties of secondary constants of uniform and uniformly varying mono-coupled periodic structures. Mechanical Systems and Signal Processing: Volume 146 (2021) 106974.