Vibrations and Acoustics

Our research in vibration and acoustics spans a broad spectrum of applications, from the structural analysis of materials to the acoustic properties of musical instruments. Through a blend of theoretical analysis, computational modelling, and experimental validation, we aim to unravel complex problems and contribute to advancements in various engineering applications.

Can a Brace be Used to Control the Frequencies of a Plate? SpringerPlus. 2:558, 2013

Dumond, P., Baddour, N.,

2013

Although many improvements in the manufacturing of guitars have been made recently, one aspect that has often been overlooked is that of the acoustical consistency of the final manufactured product. The aim of this paper is to create a better understanding of the effect of a brace on the frequencies of vibration of the brace-soundboard system. This paper seeks to shed light on why a luthier ‘tunes’ braces when a guitar soundboard is hand-manufactured. A simple analytical model of a rectangular brace and soundboard is derived from first principles using Kirchhoff plate theory in order to develop insight into the effect of the soundboard’s stiffness and brace thickness on the frequencies of the combined system. Natural frequencies and modeshapes of the combined system are calculated via the assumed shape method. Results show that by adjusting the thickness of the brace in order to compensate for the stiffness of the plate, one of the natural frequencies of the combined system can be adjusted to meet a desired value. However, simultaneously adjusting several natural frequencies cannot be done with a rectangular brace. Therefore modifications to the shape of the brace are explored.

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Effects of a Scalloped and Rectangular Brace on the Modeshapes of a Brace-Plate System. International Journal of Mechanical Engineering and Mechatronics. 1(1):1-8, 2012

Dumond, P., Baddour, N.,

2012

Shaping the soundboard braces on a wooden stringed musical instrument has long been a way in which instrument makers optimize their musical instruments. Reasons for these methods are scientifically not well understood. Various bracing patterns have successfully been used to create different-sounding wooden stringed musical instruments. These bracing patterns stimulate the modeshapes that are specific to the soundboard of the instrument. However, a higher adjustment resolution is required in order to specify the frequency spectrum of the musical instrument. This paper demonstrates how the shape of the braces affects the modeshapes of the vibrating system. A simple analytical model composed of a plate and brace is analyzed in order to see these effects. The results are plotted together for three cases: the plate by itself, the plate with a rectangular brace and the plate with a scalloped brace. For clarity, the modeshapes are analysed in 2D at different locations and along both the x and y directions of the plate. It is shown that any brace affects modeshapes for which the brace does not run along a nodal line. The different shapes of the brace are shown to affect different modeshapes by various degrees. If braces are stiffened at locations of maximum amplitude for a given modeshape, then that modeshape will be significantly affected. It is clear that by properly designing the shape of a brace, instrument makers can exert great control over the shape of the instrument's modeshapes and therefore also their frequencies.

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Effects of Using Scalloped Shape Braces on the Natural Frequencies of a Brace-Soundboard System. Applied Acoustics. 73(11):1168-1173, 2012

Dumond, P., Baddour, N.,

2012

Many prominent musical instrument makers shape their braces into a scalloped profile. Although reasons for this are not well known scientifically, many of these instrument makers attest that scalloped braces can produce superior sounding wooden musical instruments in certain situations. The aim of this paper is to determine a possible reason behind scalloped shaped braces. A simple analytical model consisting of a soundboard section and a scalloped brace is analyzed in order to see the effects that changes in the shape of the brace have on the frequency spectrum of the brace-soundboard system. The results are used to verify the feasibility of adjusting the brace thickness in order to compensate for soundboards having different stiffness in the direction perpendicular to the wood grain. It is shown that scalloping the brace allows an instrument maker to independently control the value of two natural frequencies of a combined brace-soundboard system. This is done by adjusting the brace’s base thickness in order to modify the 1st natural frequency and by adjusting the scalloped peak heights to modify the 3rd natural frequency, both of which are considered along the length of the brace. By scalloping their braces, and thus controlling the value of certain natural frequencies, musical instrument makers can improve the acoustic consistency of their instruments.

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Effects of Brace Shape on the Modeshapes of a Brace-Plate
System. Proceedings of the International Conference on Mechanical Engineering and Mechatronics. (21)1-7, 2012

Dumond, P., Baddour, N.,

2013

Shaping soundboard braces on a wooden stringed musical instrument has long been a way in which instrument makers optimize their musical instruments. Reasons for these methods are scientifically not well understood. This paper demonstrates how the shape of the braces affects the modeshapes of the vibrating system. A simple analytical model composed of a plate and brace is analyzed in order to see theses effects. The results are plotted together for three cases: the plate by itself, the plate with a rectangular brace and the plate with a scalloped brace. It is shown that any brace affects modeshapes for which the brace does not run along a nodal line. The different shapes of the brace are shown to affect different modeshapes by various degrees. If braces are stiffened at locations of maximum amplitude for a given modeshape, then that modeshape will be significantly affected. It is clear that by properly designing the shape of a brace, instrument makers can exert great control over the shape of the instrument's modeshapes and therefore also their frequencies.

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Comparison of the Assumed Shape and Finite Element Methods
for a Vibrating Brace-Plate System. Proceedings of the CSME Forum. (211)1-6, 2010

Dumond, P., Baddour, N.,

2010

Although many would agree that the finite element method is the method of choice when modeling vibrating systems, new symbolic computational tools have made it possible to accurately model systems using other discrete methods that have become nothing more than methods of academic interest. This has revived interest in these methods. One such method is that of the assumed shape method. This work compares simplified models of stringed instrument soundboards built using both the finite element and assumed shape methods. The same model is built using both methods using typical approximations when necessary. Results demonstrate that similar results are obtained using both methods. The finite element method requires a large number of elements in order to converge to decent results, but because it is solved numerically, it remains computationally less cumbersome then the assumed shape method. The assumed shape method uses much fewer functions in order to converge to reasonable results. It is also much easier to see modeling mistakes because of its intuitive approach and transparent solution methods.

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Towards Improving the Manufactured Consistency of Wooden
Musical Instruments Through Frequency Matching. Transactions of NAMRI/SME. 38:245
252, 2010

Dumond, P., Baddour, N.,

2010

Although many improvements in the manufacturing of musical instruments have been made recently, one aspect that has often been overlooked is that of the acoustic consistency of the final manufactured product. The aim of this work is to create a method in which a soundboard can be frequency matched to a brace in order to meet a set standard after assembly. A simple analytical model is created in order to study the effect of the plate's stiffness and brace thickness on the combined system. The assumed shape method is used in the analysis. Results show that by adjusting the thickness of the brace in order to compensate for the stiffness of the plate, one of the natural frequencies can be adjusted to meet a certain value. However, matching multiple natural frequencies cannot be done with a rectangular brace. Therefore modifications to the shape of the brace are suggested.

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University of Ottawa Rolling-element Dataset – Vibration and Acoustic Faults under Constant Load and Speed conditions – Spectrograms (UORED-VAFCLS-P1).

*Sehri, M., Dumond, P.

2023

Each set of raw data is obtained at a constant nominal rotational speed and load. Four fault types are collected: inner race, outer race, ball, and cage (five of each type). A total of 20 bearings are tested, where each bearing has three states of data collection: healthy, developing fault, and faulty. As a result, 60 distinct sets of data are included. For all cases, data is collected at a sampling rate of 42,000 Hz for a total 10 seconds per set of data. Raw data is provided in the folders as comma separated values files (.csv), Excel files (.xlsx), and MATLAB files (.mat) Spectrograms are created from the raw data by using the short-time Fourier transform (STFT), while using a Hanning window to convert the accelerometer and microphone values into 2D images. For each raw data file, 400 spectrogram images are created with a signal length of 512. A total of 24,000 spectrogram images are created, 12,000 images for each of the accelerometer and microphone data files. The processed data are provided in the zip files as portable network graphics files (.png).

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University of Ottawa Rolling-element Dataset – Vibration and Acoustic Faults under Constant Load and Speed conditions (UORED-VAFCLS).

*Sehri, M., Dumond, P.

2023

Spectrograms are created from the raw data by using the short-time Fourier transform (STFT), while using a Hanning window to convert the accelerometer and microphone values into 2D images. For each raw data file, 400 spectrogram images are created with a signal length of 512. A total of 24,000 spectrogram images are created, 12,000 images for each of the accelerometer and microphone data files. The processed data are provided in the zip files as portable network graphics files (.png).

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Structured Approaches to General Inverse Eigenvalue Problems, in V. N. Katsikis (ed), Applied Linear Algebra in Action (InTech, 2016) pp. 27-55

Dumond, P., Baddour, N.,

2016

An inverse eigenvalue problem is one where a set or subset of (generalized) eigenvalues is specified and the matrices that generate it are sought. Many methods for solving inverse eigenvalue problems are only applicable to matrices of a specific type. In this chapter, two recently proposed methods for structured (direct) solutions of inverse eigenvalue problems are presented. The presented methods are not restricted to matrices of a specific type and are thus applicable to matrices of all types. For the first method, the Cayley–Hamilton theorem is developed for the generalized eigenvalue vibration problem. For a given (desired) frequency spectrum, many solutions are possible. Hence, a discussion of the required information and suggestions for including structural constraints are given. An algorithm for solving the inverse eigenvalue problem using the generalized Cayley–Hamilton theorem is then demonstrated. An algorithm for solving partially described systems is also specified. The Cayley–Hamilton theorem algorithm is shown to be a good tool for solving inverse generalized eigenvalue problems. Examples of application of the method are given. A second method, referred to as the inverse eigenvalue determinant method, is also introduced. This method provides another direct approach to the reconstruction of the matrices of the generalized eigenvalue problem, given knowledge of its eigenvalues and various physical parameters. As for the first method, there are no restrictions on the type of matrices allowed for the inverse problem. Examples of application of the method are also given, including application-oriented examples.

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