# Helicopter Rotor Thickness Noise Control Using Unsteady Force Excitation
## 2 Principle of Thickness Noise Reduction  
In theoretical study, the formulation F1A (1992), an integral form of the FW-H equation, is employed for noise prediction. The noise prediction solver was validated in reference (1992), thus only a brief description is given here. The acoustic pressure in the F1A formulation can be expressed as a summation of thickness noise \(p_{T}^{\prime}\) and loading noise \(p_{L}^{\prime}\):  
\[p^{\prime}(\overset{\rightarrow}{x},t)=p_{T}^{\prime}(\overset{\rightarrow}{ x},t)+p_{L}^{\prime}(\overset{\rightarrow}{x},t) \tag{1}\]  
where,  
\[4\pi p_{T}^{\prime}(\overset{\rightarrow}{x},t)=\int\limits_{f=0}\left[\frac {\rho_{0}v_{n}}{r(1-M_{r})^{2}}+\frac{\rho_{0}v_{n}M_{r}}{r(1-M_{r})^{3}}+\frac {\rho_{0}v_{n}a_{0}(M_{r}-M^{2}))}{r^{2}(1-M_{r})^{3}}\right]_{ret}dS \tag{2}\]  
\[4\pi p_{F}^{\prime}(\overset{\rightarrow}{x},t)=\int\limits_{f=0}\left[\frac {F}{a_{0}r(1-M_{r})^{2}}+\frac{FM_{r}}{a_{0}r(1-M_{r})^{3}}+\frac{F-F_{M}}{r^{2 }(1-M_{r})^{2}}+\frac{F(M_{r}-M^{2})}{r^{2}(1-M_{r})^{3}}\right]_{ret}dS \tag{3}\]\(\rho_{0},a_{0}\) are density and sound speed in undisturbed air, \(v_{n}\) is the local normal velocity of blade surface, \(r\) is the length of the radiation vector, \(M\) is the sectional Mach number, and \(F\) is the force on the blade surface. The subscript \(r\) denotes the radiation direction (e.g., \(F_{r}=\overset{\rightarrow}{F}\cdot\overset{\rightarrow}{r}\)). The superscript \((. )\) denotes the rate of variation with respect to the source time. The subscript _ret_ indicates that the integrals are evaluated at the retarded time. The terms with \(r^{2}\) are inversely proportional to the square of distance and can be neglected when the observer is far away from the rotor. Then, Equations (2) and (3) can be rewritten as:  
\[4\pi p_{T}^{\prime}(\overset{\rightarrow}{x},t)=\int\limits_{f=0}\ \left[ \begin{array}{c} \overset{\text{velocity time derivative term}}{\overset{\rho_{0}v_{n}}{r(1-M_{r})^{2}}} \quad\quad+\overset{\text{velocity term}}{\overset{\rho_{0}v_{n}M_{r}}{r(1-M_{ r})^{3}}}\end{array} \right]\ dS \tag{4}\]  
\[4\pi p_{F}^{\prime}(\overset{\rightarrow}{x},t)=\int\limits_{f=0}\ \left[ \begin{array}{c} \overset{\text{force time derivative term}}{\overset{\text{F}}{a_{0}r(1-M_{r})^{2}}} \quad\quad+\overset{\text{force term}}{\overset{\text{F}}{a_{0}r(1-M_{r})^{3}} }\end{array} \right]\ dS \tag{5}\]  
The principle of thickness noise reduction can be explained by the above two equations. To achieve it, a 2-m diameter model rotor was used to conduct analytical research. The rotor parameters and operation conditions are given in Section 3.1. The rotor thickness noise is caused by the periodic displacement of air produced by blades rotating. According to Equation (4), the sound pressure comes from the contributions of oscillation of airflow velocity (velocity term) and its time derivative (time derivative term). Figure 1 shows the time histories of overall sound pressure and contributions from two terms at a far-field observation position (100 times radius). The positive sound pressure waveform is generated by velocity term, while the negative pressure, which determines the main waveform feature of the thickness noise, is caused by the time derivative term. Equation (5) governs the noise generated from air-load and has consistent form with thickness noise Equation (4). Through adjusting the amplitude and direction of force, the thickness noise can be partly canceled. This is the theoretical foundation of the concept of sound cancelation control. In this paper, a thickness noise control method was developed based on the unsteady on-blade force. As shown in Figure 2, the unsteady aerodynamic force was arranged at the outer portion of the blade to excite a controllable sound wave (anti-noise) opposite to the original noise (thickness noise). The superposition of them could cancel the sound pressure in a certain area, thereby reducing noise. The thickness noise radiated along the rotor plane and decreased gradually out of plane (as seen in Figure 1).