Vibrations and Acoustics

Ribbed plates are common components in structures, aircrafts, and ships. Owing to safety considerations and proper functioning, it is important to predict the natural frequencies and mode-shapes of such systems efficiently. There are many analytical, numerical, and experimental methods that extract the modal parameters of ribbed plates. However, each present advantages and tradeoffs. The aim of this paper is to implement various modeling and experimental approaches, optimize them, and compare their strengths and weaknesses. Analytical models are based on the assumed-modes method and differ in geometric configuration and trial function selection. Finite element models employ higher-order shear theories. Two experiments are performed to compare theoretical results, and to weigh the benefits of the two common setups. Parallelization and concurrency, mathematical simplifications, and algorithmic improvements are used to optimize the performance of the analytical models so that the benefits of each geometry can be understood. Parallelized programs are then put through algorithmic analysis to estimate local order of growth and study running time. By comparing the performance, accuracy, simplicity, stability, affordability, and parametrization potential of the approaches, vibration scientists and engineers are better able to select methods suitable for their research, application, or design.

The experimental study of vibrating plates having simply-supported boundary conditions can be difficult to achieve due to the complexity of preventing translation, but allowing rotation along all boundaries simultaneously. Only a few methods have been proposed, but all are either time-consuming to set up and involve customization of the test rig for each plate or do not allow the plate to be reused for other purposes. The method described in this paper offers a low-cost, simple, accurate and non-destructive way of experimentally measuring the modal properties of thin, simply supported plates and can be used for quick validations of models and designs without modification for multiple trials and varying plate properties. The key attributes of this method include: •An adjustable sliding support frame which can be made of a distinct material from the plate and which can accommodate variations in plate geometry and properties without modification.•Removable flexible sealant applied in a v-groove on the supporting frame which can be easily used to fix and support the plate according to the simply-supported boundary conditions.•A low-profile design, which can be used to accommodate most experimental testing methods for determining modal properties of vibrating plates.

Design-for-frequency of mechanical systems has long been a practice of iterative procedures in order to construct systems having desired natural frequencies. Especially problematic is achieving acoustic consistency in systems using natural materials such as wood. Inverse eigenvalue problem theory seeks to rectify these shortfalls by creating system matrices of the mechanical systems directly from the desired natural frequencies. In this paper, the Cayley–Hamilton and determinant methods for solving inverse eigenvalue problems are applied to the problem of the scalloped braced plate. Both methods are shown to be effective tools in calculating the dimensions of the brace necessary for achieving a desired fundamental natural frequency and one of its higher partials. These methods use the physical parameters and mechanical properties of the material in order to frame the discrete problem in contrast to standard approaches that specify the structure of the matrix itself. They also demonstrate the ability to find multiple solutions to the same problem. The determinant method is found to be computationally more efficient for partially described inverse problems due to the reduced number of equations and parameters that need to be solved. The two methods show great promise for techniques which could lead to the design of complex mechanical systems, including musical instrument soundboards, directly from knowledge of the desired natural frequencies.

Wood remains the material of choice for stringed musical instruments. However, as supplies of suitable wood decrease, a need to maximize the use of available timber arises. Since large variations exist within a species, manufacturing processes require direct measurements of the mechanical properties of a given specimen in order to construct acoustically consistent instruments. This paper reports on the application of non-destructive mechanical property tests developed to be used on a single wooden specimen. In this way, relationships between mechanical properties of clear, straight-grained, quartersawn timber can be investigated and the number of measurements during manufacturing can be reduced. Measured properties include density, moisture content, radial and longitudinal Young’s moduli, shear modulus and Poisson’s ratios in the longitudinal–radial plane. Furthermore, natural frequencies of theses specimens have been measured for the rectangular plates having simply supported boundary conditions. Sitka spruce (Picea sitchensis) has been studied. Relationships between various mechanical properties are found, and suggestions for using Poisson’s ratios are proposed. These relationships are shown to reduce the number of measurements required to construct acoustically consistent instruments in musical instrument manufacturing. In particular, it is shown that measurements of the Young’s moduli in both the radial and longitudinal directions would be suitable for describing the important mechanical properties of such soundboard wood. The natural frequencies of the tested plates were also calculated from the measured properties and compared to values obtained experimentally. Great correlation is found between the theoretical model and experimental results for the simply supported rectangular plates.

An inverse eigenvalue problem approach to system design is considered. The Cayley-Hamilton theorem is developed for the general case involving the generalized eigenvalue vibration problem. Since many solutions exist for a desired frequency spectrum, a discussion of the required design information and suggestions for including structural constraints are given. An algorithm for solving the inverse eigenvalue design problem using the generalized Cayley-Hamilton theorem is proposed. A method for solving partially described systems is also specified. The Cayley-Hamilton theorem algorithm is shown to be a good design tool for solving inverse eigenvalue problems of mechanical and structural systems.