8 mm, mass density 812.5 kg/m3, and dimensions a × b = 0.7 × 0.6 m2.
Solution
The plywood board has properties E = 6 × 109 N/m2, S = a × b = 0.7 × 0.6 m2, and ν = 0.3. The mass of the plate per unit surface area ρs = ρ × h = 812.5 × 0.008 = 6.5 kg/m2. and the bending stiffness B = Eh3/12(1 − ν2) = 6 × 109(0.008)3/12 (1 − [0.3]2) = 281.3 N/m. Then, the fundamental mode (the lowest natural frequency) is given for m = n = 1. Replacing the values in Eq. (2.70a) yields
b) Forced Vibration of a Rectangular Plate
Let us assume that a distributed harmonic force acts on a rectangular plate (referred to unit plate area)
(2.72)
Under the influence of this force, a plate deflection W is produced with the distribution
(2.73)
The distribution of the deflection amplitude W0 can be represented by the resultant of a series of sinusoidal vibrations. The frequency is equal to the frequency of free vibration of the plate, that is to say, the following equation must be satisfied [18]
(2.74)
where kmn = ωmn/cb is the wavenumber of the (m,n) mode of free vibration, cb is the velocity of bending waves in the plate and Ψmn are the dimensionless coefficients which determine the relative distribution in relation to the maximum amplitude of deflection.
Using the orthogonality of the function Ψmn and Fourier series, the resultant distribution of vibration on the rectangular plate is given by [16]
(2.75)
where (x′, y′) denotes the coordinates of the position of the force f (referred to unit plate area), with respect to which the integration is carried out over the surface area of the plate. The ωmn are real numbers for a plate without losses. For the frequency of forced vibration ω = ωmn, the amplitude of the deflection grows theoretically to infinity. In a plate with losses, ωmn are complex numbers, as discussed in Section 2.4.3.
For example, for a simply‐supported rectangular plate, the distribution Ψmn (see Eq. (2.69)) is
(2.76)
and γ = 1/4. Then, we can obtain that for a point force F concentrated at the point (x0, y0)
(2.76)
The vibration velocity distribution on the rectangular plate can be obtained as
(2.77)
In practice, to calculate the plate response to a concentrated force, it is necessary to truncate the infinite summation. More complicated cases of boundary conditions have been studied in the literature [19].
All the theory presented above and the subsequent applications have been developed assuming light fluid loading, so that the plate response is not affected by the surrounding environment, which acts as added mass and radiation damping [20]. This criterion is not valid for submerged structures.
Example 2.13
The numerical results for the velocity level at the coordinate point (x,y) = (0.04, 0.03) for a plywood board with properties E = 6 × 109 N/m2, ρS = 6.5 kg/m2, a × b = 0.7 × 0.6 m2, ν = 0.293 and η = 0.01 are presented in Figure 2.19. The reference velocity is 10−18 (m/s)2. It is observed that excitation close to a corner excites more modes in the plate. Excitation at the middle point of the plate excites mostly the modes (m,n) where m and n are both even, while no contribution to the velocity level is included from modes having nodal lines passing through the center of the plate.
Figure 2.19 Computed velocity level of a simply‐supported rectangular plate excited: ______ at a corner; _ _ _ _ at the center.
(Source: from Ref. [16].)
References
1 1 Bies, D.A. and Hansen, C.H. (2003). Engineering Noise Control–Theory and Practice, 3e. London: E & FN Spon.
2 2 Bell, L.H. (1982). Industrial Noise Control – Fundamentals and Applications. New York: Marcel Decker.
3 3 Hall, D.E. (1987). Basic Acoustics. New York: Wiley.
4 4 Kinsler, L.E., Frey, A.R., Coppens, A.B., and Sanders, J.V. (1999). Fundamentals of Acoustics, 4e. New York: Wiley.
5 5 Fahy, F.J. and Walker, J.G. (eds.) (1998). Fundamentals of Noise and Vibration. London: E & FN Spon.
6 6 Lighthill, M.J. (1978). Waves in Fluids. Cambridge: Cambridge University Press.
7 7 Pierce, A.D. (1981). Acoustics: An Introduction to Its Physical Properties and Applications. New York: McGraw‐Hill (reprinted by the Acoustical Society of America, 1989).
8 8 Inman, D.J. (2007). Passive damping. In: Handbook of Noise and Vibration Control (ed. M.J. Crocker), 225–231. New York: Wiley.
9 9 Ungar, E.E. (2007). Damping of structures and use of damping materials. In: Handbook of Noise and Vibration Control (ed. M.J. Crocker), 734–744. New York: Wiley.
10 10 Shabana, A.A. (1997). Vibration of Discrete and Continuous Systems, 2e. New York: Springer‐Verlag.
11 11 Meirovitch, L. (2001). Fundamentals of Vibrations. New York: McGraw‐Hill.
12 12 Lalor, N. (1998). Fundamentals of vibration. In: Fundamentals
