Engineering Acoustics. Malcolm J. Crocker

(2.70a)

where ρ_s is the mass of the plate per unit surface area. Note that Eq. (2.70a) can be approximated by

      (2.70b)

      where f_m,n is the characteristic modal frequency (Hz) and c_L is the longitudinal wave speed in the plate material (m/s).

Figure 2.18 First six modes of a rectangular plate.

      Thus, the total solution for the free vibration of a simply‐supported plate is

      (2.71)

Note that for a square plate (a = b) ω_mn = ω_nm. Therefore modes (m,n) and (n,m) have the same frequency and they are called degenerate. This fact also happens when the length or width are integer multiple of each other.

      Example 2.12

      Determine the natural frequency of the fundamental mode of a plywood board of thickness 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 ρ_s = 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 k_mn = ω_mn/c_b is the wavenumber of the (m,n) mode of free vibration, c_b is the velocity of bending waves in the plate and Ψ_mn are the dimensionless coefficients which determine the relative distribution in relation to the maximum amplitude of deflection.

      Using the orthogonality of the function Ψ_mn 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 ω_mn are real numbers for a plate without losses. For the frequency of forced vibration ω = ω_mn, the amplitude of the deflection grows theoretically to infinity. In a plate with losses, ω_mn are complex numbers, as discussed in Section 2.4.3.

      For example, for a simply‐supported rectangular plate, the distribution Ψ_mn (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, ρ_S = 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