Piano soundboard: structural behavior, numerical and experimental study in the modal range

The low frequency broadband vibrational behavior of a piano soundboard is considered. Attention is focused on the ability of finite element models and analytical models to predict precisely the behavior of such a complicated structure—especially with its orthotropy and rib effects. In order to validate these abilities, an experimental modal analysis, considered as the reference, is compared first with a numerical calculation and then with an analytical modeling of the modal basis of the same soundboard. The high structural complexity of the soundboard exceeds the analytical capabilities, but agreement is very good for the numerical model, in the frequency domain, and equally in the spatial one. The final aim is to generate a numerical tool for designing and optimizing piano soundboards.     

Numerical simulation of a piano soundboard under downbearing

A finite element model of a piano soundboard is used to study the effect of the strings tension (downbearing) on its vibration, considering the ribs, the bridges and the crown. The downbearing is modeled with the prestress theory. Prestress calculation with linear and nonlinear models including geometric rigidity are compared in terms of the modal frequencies. The effects of the downbearing in modal frequencies and mobility are investigated and the importance of the crown on these results is evaluated. A simple phenomenological law is exhibited, which characterizes the evolution of eigenfrequencies with downbearing, including the initial crown.     

Modal parameters of two incomplete and complete guitars differing in the bracing pattern of the soundboard

Similarities and differences in vibrational behavior of two guitars having a symmetric Torres bracing pattern and an asymmetric pattern forming a lattice on a soundboard are investigated by means of the modal analysis technique and laser Doppler vibrometry (LDV) measurements. Instruments are investigated before and after a bridge and strings assembling (i.e., they are incomplete or complete). The bracing pattern and the absence/presence of the bridge and strings have some effect on modal frequencies and mode shapes. The bracing pattern does not affect the sequence of at least first three low frequency mode shapes of incomplete/complete instruments but affects their modal frequencies. Depending on frequency, the bridge behaves either as a rigid or a flexible structure.     

Acoustics of the jarana jarocha

Some acoustical measurements on the jarana jarocha primera (first), a traditional Mexican instrument, a “one-piece” body instrument, in the coastal region of the Gulf of Mexico, are presented. Experimental mobility function, radiativity function, and some radiation patterns were obtained. Harmonic analysis and visualization techniques using both the finite element method and laser speckle-Chladni methods were made to obtain the lower deflection shapes of the soundboard. The experimental analysis using mobility function measurements of the lower resonances shows behaviour very similar to that observed in the classical guitar.     

Experimental study of AO and T1 modes of the concert harp

String instruments are usually composed of a set of strings, a soundboard, and a soundbox with sound holes, which is generally designed to increase the sound level by using the acoustic resonances of the cavity. In the case of the harp, the soundbox and especially the sound holes are primarily designed to allow access to the strings for their mounting. An experimental modal analysis, associated to measurements of the acoustic velocity in the holes, shows the importance of two particular modes labeled A0 and T1 as it was done for the guitar and the violin. Their mode shapes involve coupled motions of the soundboard's bending and of the oscillations of the air pistons located in the sound holes. The A0 mode is found above the frequency of the lowest acoustically significant structural mode T1. Thus, the instrument does not really take advantage of the soundbox resonance to increase its radiated sound in low frequencies. However, contribution of mode A0 is clearly visible in the response of the instrument, confirming the importance of the coupling between the soundboard and the cavity.     

Prestress effects on the eigenfrequencies of the soundboards: experimental results on a simplified string instrument

This paper presents an experimental study of the effects of prestresses on the vibration behavior of string instruments. These prestresses are created by gluing ribs (crowning) and tensioning string (downbearing). The effects of these prestresses were previously studied numerically for a piano soundboard by Mamou-Mani et al. [J. Acoust. Soc. Am. 123, 2401-2406 (2008)] and analytically for simplified models by Mamou-Mani et al. [Acta Acust. Unit. Acust. 95, 915-926 (2009)]. In the present study, a specified test bench is designed, including a simplified soundboard (a rectangular plate), a bridge, and a single string. The plate is subjected to in-plane and transverse loads. Vibrational eigenmodes are identified using nearfield acoustical holography (NAH) measurement. The evolution of eigenfrequencies with these specific prestresses is studied. The results show the effectiveness of NAH for this purpose and a very good qualitative concordance with previous numerical and analytical calculus.     

Some characteristics of the concert harp's acoustic radiation

The way a musical instrument radiates plays an important part in determining the instrument's sound quality. For the concert harp, the soundboard has to radiate the string's vibration over a range of 7 octaves. Despite the effort of instrument makers, this radiation is not uniform throughout this range. In a recent paper, Waltham and Kotlicki [J. Acoust. Soc. Am. 124, 1774-1780 (2008)] proposed an interesting approach for the study of the string-to-string variance based on the relationship between the string attachment position and the operating deflection shapes of the soundboard. Although the soundboard vibrational characteristics determine a large part of the instrument's radiation, it is also important to study directly its radiation to conclude on the origins of the string-to-string variation in the sound production. This is done by computing the equivalent acoustical sources on the soundboard from the far field sound radiation measured around the harp, using the acoustic imaging technique inverse frequency response function. Results show that the radiated sound depends on the correlation between these sources, and the played string's frequency and location. These equivalent sources thus determine the magnitude and directivity of each string's partial in the far field, which have consequences on the spectral balance of the perceived sound for each string.     

Application of holographic interferometry to the vibrational analysis of the harpsichord

This paper presents an original piece of research using holographic interferometry as a quantitative optical diagnostic. The object under investigation was the soundboard of a harpsichord. The results obtained show that it is possible to measure the spatial vibrational behaviour of the whole soundboard with an accuracy of better than 170 nm. Several features which characterize the vibration behaviour of the soundboard have been visualized including a discontinuity created by a crack which has resulted in a phase change of the interferometric fringes. Finally, the theoretical model response of the harpsichord soundboard is determined and compared to our holographic measurement.     

The effect of inharmonic partials on pitch of piano tones

Piano tones have partials whose frequencies are sharp relative to harmonic values. A listening test was conducted to determine the effect of inharmonicity on pitch for piano tones in the lowest three octaves of a piano. Nine real tones from the lowest three octaves of a piano were analyzed to obtain frequencies, relative amplitudes, and decay rates of their partials. Synthetic inharmonic tones were produced from these results. Synthetic harmonic tones, each with a twelfth of a semitone increase in the fundamental, were also produced. A jury of 21 listeners matched the pitch of each synthetic inharmonic tone to one of the synthetic harmonic tones. The effect of the inharmonicity on pitch was determined from an average of the listeners' results. For the nine synthetic piano tones studied, pitch increase ranged from approximately two and a half semitones at low fundamental frequencies to an eighth of a semitone at higher fundamental frequencies.     

Rapid scanning autocorrelator for measurements of picosecond laser pulses

A rapid scanning autocorrelation interferometer for measurements of picosecond laser pulses is described which uses a rotating prism as scanning device in one arm of the interferometer to permit continuous display of autocorrelation traces at audio frequencies on an oscilloscope. Scan widths of more than 500 ps with high linearity can be achieved. Autocorrelation measurements of picosecond pulses from a synchronously pumped mode-locked dye laser are presented.     

Fiber optic sensor based on a sagnac interferometer including a recirculating-ring delay line with an edfa

We present a theoretical and experimental investigation of a Sagnac interferometer incorporating a fiber optic recirculating-ring delay line with an erbium-doped fiber amplifier to increase the effective length of the Sagnac loop and thereby improve the low-frequency response. Theoretical calculations show that the low-frequency response of the interferometer is enhanced as expected. However, the noise penalty of using a fiber amplifier in the ring is quite high, especially at low frequencies. The signal-to-noise ratio at low frequencies, using a superfluorescent erbium fiber source, is demonstrated as increasing by a factor of 2 compared to a single-loop Sagnac interferometer with the same total length of fiber, but without fiber amplifier.     

Stochastic interval analysis of natural frequency and mode shape of structures with uncertainties

In this paper, natural frequencies and mode shapes of structures with mixed random and interval parameters are investigated by using a hybrid stochastic and interval approach. Expressions for the mean value and variance of natural frequencies and mode shapes are derived by using perturbation method and random interval moment method. The bounds of these probabilistic characteristics are then determined by interval arithmetic. Two examples are given first to illustrate the feasibility of the presented method and the results are verified by Monte Carlo Simulations. The presented approach is also applicable to solve pure random and pure interval problems. This capability is demonstrated in the third and fourth examples through the comparisons with the peer research outcomes.     

A note on vibration of a cantilever plate immersed in water

The added mass of the fluid surrounding it plats an important role in the dynamic behaviour of a submerged structure. The first few mode shapes and the respective natural frequencies of a submerged cantilever plate are found by using a finite element procedure, eigenvalues being obtained by a simultaneous iteration technique. The influence of the water depth below the plate and also of the water's lateral extent is considered, in order to test the convergency of the results. Results on the effects of the depth of immersion on the natural frequencies and mode shapes of the cantilever plate for different aspect ratios are presented.     

FREE VIBRATION ANALYSIS OF COMPOSITE RECTANGULAR MEMBRANES WITH AN OBLIQUE INTERFACE

The natural frequencies and mode shapes of a composite rectangular membrane with no exact solutions are found by using an analytical method appropriate for the geometric feature of the title problem membrane presented here. The method has a key feature in which the theoretical development is very simple and only a small amount of numerical calculation is required. Example studies show that the natural frequencies and their associated modes obtained from the method are found to be very accurate compared with the results by the FEM (SYSNOISE) or exact solutions. Furthermore, the natural frequencies converge rapidly and accurately to the exact values or the numerical results obtained from the finite element model using meshes sufficient to yield already converging natural frequencies, even when a small number of series functions are used in the proposed method.     

A distributed-mass approach for dynamic analysis of Bernoulli-Euler plane frames

The exact dynamic analysis of plane frames should consider the effect of mass distribution in beam elements, which can be achieved by using the dynamic stiffness method. Solving for the natural frequencies and mode shapes from the dynamic stiffness matrix is a nonlinear eigenproblem. The Wittrick-Williams algorithm is a reliable tool to identify the natural frequencies. A deflated matrix method to determine the mode shapes is presented. The dynamic stiffness matrix may create some null modes in which the joints of beam elements have null deformation. Adding an interior node at the middle of beam elements can eliminate the null modes of flexural vibration, but does not eliminate the null modes of axial vibration. A force equilibrium approach to solve for the null modes of axial vibration is presented. Orthogonal conditions of vibration modes in the Bernoulli-Euler plane frames, which are required in solving the transient response, are theoretically derived. The decoupling process for the vibration modes of the same natural frequency is also presented.     

A displacement measurement system, utilizing a Wollaston interferometer

A dual interferometric displacement measurement system is presented where a Wollaston prism interferometer is employed in conjunction with a normal Michelson interferometer. The system operates without the use of external polarizers, apart from those associated with the Wollaston prism interferometer itself. It is shown that an optical path difference induced in the Michelson interferometer can be detected using the Wollaston prism in a normal interferometer arrangement. Further, the interference pattern produced by the Wollaston prism interferometer changes in a measurable, linear fashion as the optical path difference from the Michelson interferometer alters. A simple theoretical analysis of the system is presented and used to derive a computer model of the optical arrangement. Results from an experimental implementation of the system, using a Wollaston prism with a beam separation of 0.5 degrees and a superluminescent diode, of wavelength 825 nm, as a light source, are included and compared to the results from the computer model.     

Free transverse vibrations of an elastically connected rectangular plate-membrane complex system

This paper theoretically analyzes undamped free transverse vibrations of an elastically connected rectangular plate-membrane system. Solutions of the problem are formulated by using the Navier method. Natural frequencies of the system in the form of two infinite sequences are determined. Normal mode shapes of vibration expressing two kinds of vibration, synchronous and asynchronous, are presented. The initial-value problem is also solved. In a numerical example, the effect of membrane tension on the natural frequencies of this mixed system is discussed.     

Dynamic ground compliance in multistorey buildings

A study is presented of the changes in the characteristics of the natural modes of vibration for multistorey structures which are founded on flexible foundations. First a standard eigenvalue problem is formulated for the proportionally damped case. Then general relationships of changes in natural frequencies and mode shapes are derived for the linear vibration theory. By means of an example problem it is demonstrated, however, that only the first mode obeys the predicted changes of frequencies and mode shapes over a wide range of foundation stiffness. The higher modes are shown to deviate substantially from the linear behaviour. This deviation is ascribed to geometric changes in mode shapes.