Free Vibrations of Annular Plates Coupled With Fluids

Numerical simulation of fluid added mass effect on a francis turbine runner

“…Moreover, when comparing the response in air with the one in water, only small changes in the mode shapes can be observed. This has been carefully investigated and well verified by some previous publications [11,[30][31][32]. This has been also confirmed by the results found in this investigation when comparing the normalized maximum displacement of the different parts of the runner (Table 11).…”

“…With this assumption, the frequencies in water can be related to the frequencies in air, using the following equations [11,32]. The squares of circular frequencies of structure in air (x a ) and in water (x w ) can be expressed as…”

The Modal Low Frequency Noise of an Elastically Supported Circular Plate

“…The equation of motion takes the form of (1) where W mn (r, φ) is the mode shape of the mode (m, n); r  [0, a] is the radial variable; a is the plate radius; φ  [0, 2π] is the angular variable; m = 0, 1, 2, ... , ∞ and n = 1, 2, 3, ... , ∞ are the numbers of nodal diameters and nodal circles, respectively; A mn , C mn are unknown constants [10,11]. The orthogonality condition [10] is (4) where δ mp and δ nq are the Kronecker deltas, and the following integral has been used to obtain (5) (6) where λ mn = k mn a is the eigenvalue.…”

“…Therefore, it is necessary to examine acoustic quantities such as the radiation efficiency of such sound sources. A number of authors used some approximate methods for computing the radiation efficiency of some flat plates [1,2,3,4,5]. A few studies attempted to find some approximate formulas for radiation efficiency of a clamped circular plate applying some complete solutions to the vibration phenomena [6,7,8,9].…”

Asymptotic Formulae of the Modal Acoustic Impedance for the Asymmetric Vibrations of a Clamped Circular Plate

“…The excitation is very often asymmetric in the real vibrating systems and it is very useful to find some asymptotic and approximate formulae for the modal quantities necessary to compute the acoustic power, the acoustic pressure and the structure's vibration velocity including the fluid loading. So far, a number of such formulae have been presented for vibrating circular and rectangular membranes and plates using several approximate methods [1][2][3][4][5][6][7][8][9]. The results obtained by using the exact solutions for the free vibrations have been presented in a few studies [10][11][12].…”