Kirchhoff’s theory of thin plates is used to derive the forced vibration of an undamped, isotropic, rectangular plate, which is governed by the differential equation Williams (1999). ρ h ∂ 2 w ∂ t 2 + B ∂ 4 w ∂ x 4 + 2 ∂ 4 w ∂ x 2 ∂ y 2 + ∂ 4 w ∂ y 4 = u (1) In this equation, w ( x , y , t ) is the displacement of the plate from the reference state in the z -direction and u ( x , y , t ) is the lateral load. The variable B = E h 3 / ( 12 ( 1 − ν 2 ) ) denotes the bending stiffness of the plate, where E is the Young’s modulus of the plate, h is its thickness, and ν is Poisson’s ratio. Figure 1 shows the plate with length l x , width l y and the origin of the coordinate system in the lower left corner of the plate. An arbitrary time-varying single force f ( t ) applied at coordinates ( x 1 , y 1 ) excites the plate to vibrations, i.e., u ( x , y , t ) = δ ( x − x 1 ) δ ( y − y 1 ) f ( t ) with δ as the Dirac delta function.

The equation of motion Equation 1 is solved by modal analysis, which is based on the modal expansion of the deflection w , w x , y , t = ∑ n , m = 1 ∞ ϕ n m x , y q n m t (2) Here, ϕ n m ( x , y ) denotes the n m -th mode shape satisfying the simply supported boundary conditions Leissat (1973), ϕ n m x , y = 4 l x l y sin n π x l x sin m π y l y , n = 1,2,3 … , m = 1,2,3 … (3) This procedure yields the modal oscillator equations in terms of the modal coordinates q n m ( t ) Wrona et al. (2020), q ̈ n m + 2 ξ n m ω n m q ̇ n m + ω n m 2 q n m = 1 ρ p h ϕ n m x 1 , y 1 f t , n , m = 1,2,3 … (4) where damping has been added in the form of modal damping with modal damping ratio ξ n m to account for independent energy dissipation of each mode (a reasonable approximation for lightly damped systems). For each mode ( n , m ) the corresponding natural circular frequency is ω n m = B / ρ h n π / l x 2 + m π / l y 2 . The modal oscillator equations Equation 4 are solved by standard procedures of structural dynamics. Inserting the resulting modal coordinates into Equation 2 and deriving the deflection with respect to the time t yields the surface velocity distribution v of the plate, which governs the emitted sound radiation.

In particular, an aluminum plate with the properties given in Table 1 is considered in this study. For this plate, Figure 2 shows the first six mode shapes according to Equation 3 up to a frequency of 750 Hz. The corresponding natural frequencies f n m = ω n m / ( 2 π ) are 176.1 Hz, 248.9 Hz, 370.3 Hz, 540.3 Hz, 631.4 Hz, and 704.2 Hz.

It is assumed that the vibrating plate is the only sound source and that the sound waves propagate in the positive direction of the z -axis (see Figure 1). Furthermore, the viscous interaction between the vibrating structure and the surrounding air is neglected. Since the viscous boundary layer is thin compared to the acoustic wavelength at common acoustic frequencies, the viscous shear stresses and energy dissipation are negligible when compared to the forces due to pressure and inertia. The air medium is therefore treated as an inviscid fluid.

Once the surface vibration velocity distribution of the plate has been computed according to Section 2.1.1, the sound pressure p at a given point in space r = ( x , y , z ) in the frequency domain ( ω in rad/s) can be determined by the Rayleigh integral Williams (1999), Fahy and Gardonio (2007). p x , y , z , ω = i ω ρ a 2 π ∫ S v x ′ , y ′ , 0 , ω g R , ω d S (5) with g R , ω = e − i k R R (6) Here, r S = ( x ′ , y ′ , 0 ) is the position vector of the elemental surface d S , v ( x ′ , y ′ , 0 , ω ) is the normal velocity at this element, and R = | r − r s | . Moreover, ρ a is the density of air, k = ω / c is the wave number (where c is the speed of sound), and i is the imaginary unit. The parameters used for the analysis of the acoustic radiation are given in Table 1.

For the computation of the sound pressure field, the vibrating structure is discretized into elemental vibration sources Fahy (1989), Elliott and Johnson (1993). Here, elemental radiators based on a linearly spaced grid with an area A e = 0.02 × 0.02 m are chosen, which yields reliable results for frequencies up to 1 kHz. Let N be the number of elemental radiators and v = [ v 1 v 2 … v N ] T a column vector of complex amplitudes of the velocities at the surface of the vibrating plate. Then, the column vector of the sound pressure p = [ p 1 p 2 … p N ] T at a plane parallel to the plate is composed by elements of the form p q x q , y q , z q , ω = i ω ρ a A e e − i k R q j 2 π R q j v j x ′ , y ′ , 0 , ω (7) where R q j is the distance between the centers of the q th and j th elements. Since each elemental source contributes to the sound pressure at a single point in space, this procedure can be computationally expensive. The literature typically considers farfield conditions Baumann et al. (1992), where this equation is valid and can be further simplified. Since this study is aimed at room acoustics where typical nearfield conditions prevail Williams (1999), no simplification of Equation 5 can be used.

Since sound pressure is a field that varies with the observed point in space, it is common in acoustic studies to make use of the radiated sound power Fahy and Gardonio (2007) as position independent quantity. The time-averaged sound power W is obtained by integrating the product of the surface velocity v ( x ′ , y ′ , 0 , ω ) and the surface sound pressure p ( x , y , 0 , ω ) over the panel, providing an approximation for the computation of sound power in nearfield conditions. This is given by W ω = 1 2 ∫ S Re v x ′ , y ′ , 0 , ω * p x , y , 0 , ω d S (8) where the superscript ^* stands for the complex conjugate.

For the elemental radiators, W in Equation 8 is taken to be the sum of the radiated sound power of each element such that W ω = l x l y 2 N Re v H p (9) for a grid with element dimensions smaller than both the structural and acoustic wavelength, where H indicates the Hermitian transpose (transpose and conjugate).

The radiated sound power can alternatively be computed by a set of radiation modes, a concept that is widely used in ASAC approaches Johnson and Elliott (1995), Elliott and Johnson (1993). These radiation modes are frequency-dependent, but independent from each other. Through them, radiation efficiencies can be computed, measures of how well the vibrating system produces sound for each of these modes. This approach is not followed here and is left for future studies.

To account for the effects of reflections from walls around the plate, the image source method is used Nagy et al. (2006). For each elemental radiator, the reflection is taken as an image source, where its distance is taken by mirroring this vibrating element with respect to the reflecting wall. Here, first order image sources are taken into account. Equation 6 must therefore be modified according to g R 0 , R , ω = e − i k R 0 R 0 + γ 1 ω e − i k R 1 R 1 + ⋯ + γ j ω e − i k R j R j (10) where R j = | r − r j | ( r j denotes the position of the j th image source) and γ j ( ω ) represents absorption effects. Figure 3 shows an example of this method for the present system, where reflections in a room are taken into account. A generic point “ × ” is marked on the surface of the plate to illustrate this method.

Figures 8–10 show the Bode plots of the frequency response function P i / F j with respect to the target point of control and the disturbance for uncontrolled and controlled systems at a perpendicular distance z = 0.5 m from the plate. They illustrate the three excitation force location scenarios described above.

A comparison of Figure 5 with Figure 8 shows that for all the control approaches there is a significant reduction in the magnitude of the sound pressure in the considered frequency range for this case of disturbance and control point at the same position on the plate. However, for a small deviation between these two points (Figures 6, 9) it is already noticeable that vibration reduction does not necessarily imply in sound pressure reduction for all the control approaches, as seen in Figure 9B in the frequency range between 240 Hz and 380 Hz. As the distance between the excitation force and control point increases, this is also true for DVF, as demonstrated in Figure 10. Also, the PID and I control approaches, which have a better performance in terms of structural vibration mitigation (see Figure 7), have a worse performance in terms of sound pressure propagation for frequencies between the natural frequencies of the uncontrolled plate.

To evaluate if this reduction/amplification of the sound pressure occurs for the whole x − y plane at z = 0.5 m and not only for a specific point Figure 11 shows the magnitude | P i / F 1 | (for each i th element) in this plane for two frequencies. Only the uncontrolled, PID and DVF ( g P = 7.0 e + 02 ) cases are shown for the case of ( x 1 , y 1 ) = ( 0.3 , 0.24 ) and ( x 2 , y 2 ) = ( 0.59 , 0.11 ) (remote excitation force and control point, respectively). On the left column, the first natural frequency of the plate is considered, where a reduction in magnitude with respect to the uncontrolled system can be seen for all points in the two control approaches. Meanwhile, the right column refers to the frequency at which the PID results in the first peak in Figure 10B (226.5 Hz). For the DVF control, there is a decrease in magnitude for almost all points, while for the PID case, there is an increase in magnitude for all points in the plane.

Using sound pressure directly as a measurement quantity in a feedback control setup is challenging due to its position dependency and the inherent time delay of the vibroacoustic system. The phase plots in Figures 8–10 show this delay due to the propagation of the sound wave in air. Compensation of time delay systems is an ongoing field of study and more details can be found in Wang (2012), Srivastava and Pandit (2016). In the current study, (direct) acoustic control is not considered.

Figures 12–14 show the magnitude | W / F j 2 | of the radiated sound power for the uncontrolled system and the considered controlled systems for the three positions of excitation force. The change of the damped natural frequency peaks in Figures 8–10 is not only related to the sound pressure but also to the sound power. Similar to Figures 9, 10, although there is a reduction in magnitude around the peaks of the uncontrolled system, this is not necessarily true for all frequencies. This effect is clearly seen in the cases where the structural mode shapes of the controlled system differ too much from the uncontrolled one.

The overall sound pressure level (SPL) at the observation point and the overall sound power level (SWL) for uncontrolled and controlled systems without considering reflections are also of interest. For the cases where excitation point and the control point are the same or very close, it can be seen in Figures 8, 9, 12, 13 that these overall levels decrease for all control approaches. However, this is not clear at first glance when these two points are far apart. SPL and SWL for this scenario are given in Table 2, where it is evident that the PID and I controllers result in a slight increase in both levels.

To account for reflections, a constant γ j ( ω ) = 1 in Equation 10 is used for all walls in Figure 3, i.e. complete reflections are assumed for all excitation frequencies. The considered walls are at a distance of 1 m from the plate. Figures 15–17 show the Bode plots of P 2 / F j for reflections in a closed room. Compared to Figures 8–10, there is a change in phase. The peak sound pressure at the second resonance frequency of the plate would be reduced by the reflections in the uncontrolled system. Here, DVF control results in an increase in sound pressure in the controlled systems in Figure 17 (remote excitation and control points).

Figure 18 shows the same configurations as in Figure 11, now with reflections considered. The magnitudes for all points in the plane for a vibrating frequency as the first damped natural frequency of the uncontrolled plate are reduced for both controlled approaches. However, for a frequency of 226.5 Hz this is not true for all points in the plane for the DVF approach, although the magnitude increases at the observed point.

Figures 19–21 show magnitude plots of W / F j 2 for the uncontrolled system and controlled systems with the different control approaches. Compared to the case without reflections, there is a slight decrease in magnitude at the second peak and a slight increase at the third and fourth peaks for the uncontrolled plate. At lower frequencies, a difference of about 7 dB can be observed between the uncontrolled structure with and without reflections.

The SPL at the observed point and the SWL for the system with reflections are also given in Table 2 for ( x 1 , y 1 ) = ( 0.3 , 0.24 ) and ( x 2 , y 2 ) = ( 0.59 , 0.11 ) in Figure 1. Unlike the system without reflections, the SPLs for PID and I controllers at this specific point are now lower than the uncontrolled system. However, the SWLs are still slightly higher.