A Dynamic RCS and Noise Prediction and Reduction Method of Coaxial Tilt-Rotor Aircraft Based on Phase Modulation

For tilt-rotor aircraft with coaxial rotors (coaxial rotor aircraft), reduction of radar cross section as well as acoustic noise can be essential for stealth design, and the rotation of the coaxial rotors can have an influence on noise and dynamic radar cross section (RCS) characteristics. In this paper, an approach to the prediction of both the sound pressure level (SPL) of noise and the dynamic RCS of coaxial-tilt aircraft is carried out, based on the theories of the FW-H equation, the physics optics method (PO) and the physical theory of diffraction (PTD) method. In order to deal with the rotating parts (mainly including coaxial rotors), a generated rotation matrix (GRM) is raised, aiming at giving a universal formula for the time-domain grid coordinate transformation of all kinds of rotation parts with arbitrary rotation centers and rotation axis directions. Moreover, a compass-scissors model (CSM) reflecting the phase characteristics of coaxial rotors is established, and a method of noise reduction and RCS reduction based on the phase modulation method is put forward in this paper. The simulation results show that with proper CSM parameter combinations, the reduction of noise SPL can reach approximately 3~15 dB and the reduction of dynamic RCS can reach 1.6 dBsm at most. The dynamic RCS and noise prediction and reduction method can be meaningful for the radar-acoustic stealth design of coaxial tilt-rotor aircrafts.


Introduction
With the abilities of vertical take-off and landing, hovering and high-speed cruise flight, tilt-rotor aircraft can be regarded as a combination of helicopters and fixed-wing aircraft, which makes it have a broad prospect of application in military and civil aviation [1][2][3]. The CL-84 Dynavert and V-22 "Osprey" are well known as successful examples of tiltrotor aircraft. With the development of artificial intelligence and autonomous technology, tilt-rotor aircraft are widely used in the field of unmanned aerial vehicles (UAVs) for photography tasks, recon missions, etc. Cetinsoy et al. have designed and constructed a new quad tilt-wing UAV [4]. Franchi et al. have developed a tilt-hexarotor UAV [5].
The typical construction of rotor system of tilt-rotor aircraft usually consists of single rotors, and the rotation speeds of rotors are limited by the requirement of torque balance. Coaxial rotors can solve this problem, as the torque of an independent coaxial rotor can always be 0 when the two rotors rotate at the same speed in opposite directions. Therefore, coaxial rotor aircraft have the advantage of free adjustment of rotation speeds without producing extra torques. Aircraft with coaxial rotors have broad application prospects, and research on coaxial rotors has become a hotspot in recent years [6][7][8][9]. Lv et al. have designed and assembled a coaxial tilt-rotor unmanned aerial vehicle (CTRUAV) with a novel structure and possesses two pairs of front coaxial tiltable rotors that have the advantages of balanced Coriolis force, balanced reaction torques and balanced torques caused by blade flapping on the coaxial rotors [10].
As for the military coaxial tilt-rotor for recon missions, acoustic noise and radar cross section (RCS) can be significant characteristic signals that can make the coaxial tilt-rotor F.Farassat has developed a direct calculation method of the sound field of rotating sound sources based on the FW-H equation and also developed several time-domain formulations for the solution of the FW-H equation [31]. Moreover, the Kirchhoff equation for the calculation of the sound field has been deduced, which can be written as: where p represents the total sound pressure, p T is the thickness noise pressure and p L is the loading noise pressure. c 0 is the sound speed and the subscript "ret" means the sound pressure value calculated at retarded time. According to finite element method (FEM), the geometry model of the rotor blades can be divided into triangle facet elements for the integral calculations of thickness and loading noise. Therefore, Equations (2) and (3) can be transformed into: In Equations (4) and (5), r i is determined by the position of facet element numbered i, M i is determined by the position as well as the rotation speed of facet element numbered i, and thus the thickness and loading noise of each facet element can both be regarded as functions of its position and rotation speed. Therefore, Equations (4) and (5) can be written as: (THICK(ω i , X(∆S i ))) (6) p L (x, t) = 1 4π In the hovering status and the low-speed steady flight, the aerodynamic noise mainly consists of the thickness noise and the loading noise, and therefore the Farassat formulation 1A can give a proper prediction for the noise generated by the coaxial rotors in the hovering status and the low-speed steady flight. The total noise pressure equals the linear superimposition of the thickness noise and loading noise produced by each of the rotors.

PO and PTD Method for RCS Calculation
As an effective method of RCS prediction, the PO method is applied to the prediction of RCS for coaxial tilt-rotor aircraft. According to the finite element method (FEM), the geometry model of the target can be divided into triangle elements, as shown in Figure 1,  Table 1. The PO method can be applied to the calculation of all triangular facets.
superimposition of the thickness noise and loading noise produced by each of the rotors.

PO and PTD Method for RCS Calculation
As an effective method of RCS prediction, the PO method is applied to the prediction of RCS for coaxial tilt-rotor aircraft. According to the finite element method (FEM), the geometry model of the target can be divided into triangle elements, as shown in Figure 1, and the sizes of mesh generations for different parts of the aircraft model are listed in Table 1. The PO method can be applied to the calculation of all triangular facets.    The far-field electric field can be obtained by: where E S represents the scattered electric field, r represents the field point, r represents the coordinate vector of source point. J S represents the surface current, which can be calculated by: In Equation (9), H i represents the magnet field of the incident wave. For far field, the incoming wave can be considered a plain wave, and thus the RCS of the target can be written as: Substitute Equation (8) into Equation (10), it can be obtained that Taking the relationship between the magnetic field and the electronic field, the calculation of RCS can be transferred into: In Equation (12), the integral item I can be written as: where N represents the number of sides of every mesh element. N = 3 for triangle mesh elements, which are most commonly used in mesh generation.
In order to obtain more accurate RCS results, the edge diffraction effects of the target should also be taken into consideration, and PTD is applied to the RCS calculation of the target. More details of PTD can be seen in References [28][29][30], and the total RCS can be obtained by: where elements with subscript E represent the edge elements and elements with subscript F represent the face elements.
Moreover, the essence of the time-variation of dynamic RCS can be explained as the time-variation of position of facet and edge elements, and RCS is determined by the relative positions of the elements and the observer point. Thus, the total dynamic RCS can be regarded as a function of the position of each element.

Dynamic Calculation Formulas with Generalized Rotation Matrix (GRM)
As for targets with moving parts, such as helicopters with rotating rotors, the noise and RCS values of these sorts of targets change over time. It is essential that a dynamic method be introduced for the time-domain solution of these targets. Zhou ZY has developed a method called the dynamic scattering method (DSM), which can be applied to dealing with the RCS of rotating rotors. However, the grid coordinate transformation matrixes of the DSM seem to be separated into different formulas, which leads to difficulties in comprehension and concise expressions for rotation model establishment. Moreover, the expressions of the position of the rotation center and the direction of the rotation axis in the grid coordinate transformation matrixes of DSM are rather vague. Therefore, based on the grid coordinate transformation matrixes of DSM, a concise and comprehensive matrix called generalized rotation matrix (GRM) is put forward, aiming at giving a universal formula for the time-domain grid coordinate transformation of all kinds of rotation parts with arbitrary rotation centers and rotation axis directions.
As is shown in Figure 2, the situation of an element rotating around one rotation axis is taken into consideration. The coordinate of rotation center is set as X 0 = x 0 y 0 z 0 T , and the direction vector of the rotation axis is set as n = [cos ϕ sin θ, sin ϕ sin θ, cos θ] T .
Set O x y z as the coordinate system, with z axis parallel to the rotation axis and z O x plane vertical to xOy plane. The position vector of the rotating element in O x y z is set as x y z T , and O is considered as the rotation center, whose position vector is method be introduced for the time-domain solution of these targets. Zhou ZY has developed a method called the dynamic scattering method (DSM), which can be applied to dealing with the RCS of rotating rotors. However, the grid coordinate transformation matrixes of the DSM seem to be separated into different formulas, which leads to difficulties in comprehension and concise expressions for rotation model establishment. Moreover, the expressions of the position of the rotation center and the direction of the rotation axis in the grid coordinate transformation matrixes of DSM are rather vague. Therefore, based on the grid coordinate transformation matrixes of DSM, a concise and comprehensive matrix called generalized rotation matrix (GRM) is put forward, aiming at giving a universal formula for the time-domain grid coordinate transformation of all kinds of rotation parts with arbitrary rotation centers and rotation axis directions. As is shown in Figure 2, the situation of an element rotating around one rotation axis is taken into consideration. The coordinate of rotation center is set as x y z ′ ′ ′ , and O′ is considered as the rotation center, whose position vector is The transformation relationship between the coordinates Oxyz and O x y z can be obtained as: When the element starts to rotate around the rotation axis with the angular velocity of ω, x y z T and the initial position vector x y z Therefore, the relationship between the absolute position vector x y z T and the initial O x y z position vector x y z T t=0 can be written as: At the time point of t = 0, the absolute position vector and the initial O x y z position vector also satisfy the relationship of Equation (18), as is illustrated in Equation (19): Combine Equation (18) with Equation (19), it can be obtained that the absolute position of the rotation element can be considered as a function of time, which can be written as: In Equation (20), the generated rotation matrix (GRM) can be obtained as: where From Equation (21), it can be seen that GRM contains the information of time, rotation speed and direction of the rotation axis. The rotation speed equals to 0 when the rotation element keeps still, and then indicating the stationary state of the rotation element. The whole model of coaxial tilt-rotor aircraft can be considered as consisting of rotating parts (rotors) and steady parts (fuselages), which can be written as: According to the finite element method (FEM), the parts can be written as matrix of Equation (29): The acoustic pressure of noise can be obtained by: For the calculation of dynamic RCS, the model matrix can be written as: where P Fm (t) represents the set of moving facet elements, P Em (t) represents the set of moving edge elements, P Fs (t) represents the set of static facet elements and P Es (t) represents the set of static edge elements. Therefore, the dynamic RCS can be obtained by:

Noise Prediction
For the verification of the noise prediction method put forward above, the experimental data [32] of the noise of a sort of rotor is used. The rotor is made up of two blades that are equally placed, with the airfoil of NACA0010, a chord of 0.4 m and 10 m in diameter. The speed of sound is c 0 = 340.75 m/s and the air density is ρ 0 = 1.234 kg/m 3 . The rotor runs in the case where the inflow Mach number is 0.2, parallel to the rotation axis. The inflow Mach vector can be obtained as (0, 0, 0.2). The blade tip Mach is 0.6. The distance between the observer and the rotation axis is 50 m, and the location of the observer is in the XY plane, the rotation plane.
The comparison of the sound pressure of the experimental results and the results calculated using the method is shown in Figure 3. It can be clearly illustrated from Figure 3 that the calculated results duplicate the experimental results quite well, which can prove the validity of the method for calculating acoustic noise.

Dynamic RCS Prediction
The experimental data of the dynamic RCS of a sort of tilt-rotor aircraft model [28] is applied for the verification of the dynamic RCS prediction method put forward above. The blade passing period of the rotor has been divided into time points, and different geometry models at different time points have participated in the calculation in FEKO. The dynamic RCS is calculated using the method developed above. The radar wave frequency is 5 GHz, and the rotation speed is fixed at 1600 r/min.
The comparison of the dynamic RCS curve of the experimental results and the dynamic RCS curve calculated using the method is shown in Figure 4, indicating that the calculated results fit the FEKO results quite well, which can prove the validity of the method for dynamic RCS calculation. tance between the observer and the rotation axis is 50 m, and the location of the observer is in the XY plane, the rotation plane.
The comparison of the sound pressure of the experimental results and the results calculated using the method is shown in Figure 3. It can be clearly illustrated from Figure  3 that the calculated results duplicate the experimental results quite well, which can prove the validity of the method for calculating acoustic noise.

Dynamic RCS Prediction
The experimental data of the dynamic RCS of a sort of tilt-rotor aircraft model [28] is applied for the verification of the dynamic RCS prediction method put forward above. The blade passing period of the rotor has been divided into time points, and different The dynamic RCS is calculated using the method developed above. The radar wave frequency is 5 GHz, and the rotation speed is fixed at 1600 r/min. The comparison of the dynamic RCS curve of the experimental results and the dynamic RCS curve calculated using the method is shown in Figure 4, indicating that the calculated results fit the FEKO results quite well, which can prove the validity of the method for dynamic RCS calculation.

Geometry Model of Coaxial Tilt-Rotor Aircraft
The geometry model of coaxial tilt-rotor aircraft is established according to the geometry parameters of V22 tilt-rotor aircraft, as is shown in Figure 5

Geometry Model of Coaxial Tilt-Rotor Aircraft
The geometry model of coaxial tilt-rotor aircraft is established according to the geometry parameters of V22 tilt-rotor aircraft, as is shown in Figure 5. R r represents the radius of the rotors, and d rotor represents the distance between the lower rotor and upper rotor of each coaxial rotor. d axis is the distance between the rotation axis of the two coaxial rotors. As for the fuselage, L f represents the length of the fuselage, W wing represents the wingspan of the main wings and W tail represents the wingspan of the horizontal tail. The numerical values for the geometry parameters are listed in Table 2.

Compass-Scissors Model (CSM) of Coaxial Rotors
With the two rotors rotating at the same rotation speed in opposite directions, a coaxial rotor appears to have obvious symmetry and directivity. A model called the compass-scissors model (CSM) has been put forward for the description of the orientation and rotation phase status of coaxial rotors. The descriptions of the main elements and the main parameters of CSM are as follows. CCW rotor, CW rotor and pointer blades A coaxial rotor consists of two rotors rotating in opposite directions. The rotor rotating counterclockwise is defined as a CCW rotor, while the rotor rotating clockwise is defined as a CW rotor. For each rotor of the coaxial rotor, one of the blades can be selected to be the pointer blade, and the angle formed by the pointer blade and the defined reference direction can be regarded as the direction angle of the rotor. The elements mentioned above are shown in Figure 6.

Compass-Scissors Model (CSM) of Coaxial Rotors
With the two rotors rotating at the same rotation speed in opposite directions, a coaxial rotor appears to have obvious symmetry and directivity. A model called the compassscissors model (CSM) has been put forward for the description of the orientation and rotation phase status of coaxial rotors. The descriptions of the main elements and the main parameters of CSM are as follows.
a. CCW rotor, CW rotor and pointer blades A coaxial rotor consists of two rotors rotating in opposite directions. The rotor rotating counterclockwise is defined as a CCW rotor, while the rotor rotating clockwise is defined as a CW rotor. For each rotor of the coaxial rotor, one of the blades can be selected to be the pointer blade, and the angle formed by the pointer blade and the defined reference direction can be regarded as the direction angle of the rotor. The elements mentioned above are shown in Figure 6. Compass angle and scissors angle Despite the rotation of the CCW rotor and the CW rotor (the rotor rotating clockwise), the angle bisector of the pointer blades of the two rotors remains still, just like the pointer of a compass. The angle formed by the angle bisector and the defined reference direction

b. Compass angle and scissors angle
Despite the rotation of the CCW rotor and the CW rotor (the rotor rotating clockwise), the angle bisector of the pointer blades of the two rotors remains still, just like the pointer of a compass. The angle formed by the angle bisector and the defined reference direction is defined as "compass angle", determining the orientation status of the coaxial rotor. The signal for the compass angle is set as ψ in this paper.
With the passing of time, the pointer blades rotating away from the angle bisector can form an "opening scissors". Thus, the angle formed by the angle bisector and one of the pointer blades can be defined as the "scissors angle", reflecting the rotation phase status of the coaxial rotor. Therefore, the model can be named as a compass-scissors model (CSM). The signal of the scissors angle is set as χ in this paper.
By observing the main elements and the main parameters of CSM, it can be inferred that CSM has the ability to describe the orientation-time status of coaxial rotor. The orientation status of a coaxial rotor is determined by compass angle, while the time status of a coaxial rotor is determined by the scissors angle. Obviously, coaxial rotors with the same rotation axis and rotation speed but different compass angles and different scissors angles can lead to quite different noise and RCS at the same observer point.
As for coaxial tilt-rotor aircraft, the rotor system consists of two coaxial rotors. According to the periodicity of coaxial rotors, different scissors angles of two coaxial rotors can illustrate the same situation as long as the scissors angle difference of the two coaxial rotors stays the same. For example, the situation of "left scissors angle = 90 • , right scissors angle = 110 • " represents the situation of "left scissors angle = 0 • , right scissors angle = 20 • " with time passing T/4 (T = 2π/ω), as is shown in Figure 7, and aircraft in the two situations can produce the same noise and RCS. is defined as "compass angle", determining the orientation status of the coaxial rotor. The signal for the compass angle is set as ψ in this paper.
With the passing of time, the pointer blades rotating away from the angle bisector can form an "opening scissors". Thus, the angle formed by the angle bisector and one of the pointer blades can be defined as the "scissors angle", reflecting the rotation phase status of the coaxial rotor. Therefore, the model can be named as a compass-scissors model (CSM). The signal of the scissors angle is set as χ in this paper.
By observing the main elements and the main parameters of CSM, it can be inferred that CSM has the ability to describe the orientation-time status of coaxial rotor. The orientation status of a coaxial rotor is determined by compass angle, while the time status of a coaxial rotor is determined by the scissors angle. Obviously, coaxial rotors with the same rotation axis and rotation speed but different compass angles and different scissors angles can lead to quite different noise and RCS at the same observer point.
As for coaxial tilt-rotor aircraft, the rotor system consists of two coaxial rotors. According to the periodicity of coaxial rotors, different scissors angles of two coaxial rotors can illustrate the same situation as long as the scissors angle difference of the two coaxial rotors stays the same. For example, the situation of "left scissors angle = 90°, right scissors angle = 110°" represents the situation of "left scissors angle = 0°, right scissors angle = 20°" with time passing T/4 (

/ T
π ω = ), as is shown in Figure 7, and aircraft in the two situations can produce the same noise and RCS. Therefore, a combined parameter called scissors angle difference is set up for coaxial tilt-rotor aircraft, reflecting the phase difference of the two coaxial rotors. The scissors angle difference is defined as:

Simulation
In this paper, simulations of coaxial tilt-rotor aircraft in helicopter mode and fixedwing mode are carried out for the exploration of the noise and dynamic RCS characteristics. In the simulation cases shown in Figures 8 and 9, the detector receives both acoustic noise and scattered radar waves from coaxial tilt-rotor aircraft at the elevation angle β = 45° and the azimuth angle at α =0°(forehead), 90°(side) and 180°(rear). The distance Therefore, a combined parameter called scissors angle difference is set up for coaxial tilt-rotor aircraft, reflecting the phase difference of the two coaxial rotors. The scissors angle difference is defined as:

Simulation
In this paper, simulations of coaxial tilt-rotor aircraft in helicopter mode and fixedwing mode are carried out for the exploration of the noise and dynamic RCS characteristics. In the simulation cases shown in Figures 8 and 9, the detector receives both acoustic noise and scattered radar waves from coaxial tilt-rotor aircraft at the elevation angle β = 45 • and the azimuth angle at α = 0 • (forehead), 90 • (side) and 180 • (rear). The distance from the helicopter to the detector is 1000 m. In helicopter mode, the speed of the aircraft is set to 0, illustrating the status of hovering. In fixed-wing mode, the speed of the aircraft is set to 139 m/s (500 km/h), illustrating the status of high speed forward flight. from the helicopter to the detector is 1000 m. In helicopter mode, the speed of the aircraft is set to 0, illustrating the status of hovering. In fixed-wing mode, the speed of the aircraft is set to 139 m/s (500 km/h), illustrating the status of high speed forward flight.    from the helicopter to the detector is 1000 m. In helicopter mode, the speed of the aircraft is set to 0, illustrating the status of hovering. In fixed-wing mode, the speed of the aircraft is set to 139 m/s (500 km/h), illustrating the status of high speed forward flight.

Results and Discussion
As for the results of previous research, only a few conditions have been taken into consideration, and the influences of rotor phase parameters on RCS and noise SPL cannot be observed entirely. Therefore, carpet diagrams showing the distribution of RCS and noise SPL under different CSM parameter combinations are introduced in this paper, and the optimized CSM parameter combinations (design points) can be selected from the carpet diagrams.
In this paper, 12 cases are considered for the calculation of the noise SPL and dynamic RCS of coaxial tilt-rotor aircraft at different azimuth angles and flight modes. Six of the cases are carried out for noise SPL calculation, while the other six of the cases are for dynamic RCS calculation. The various parameters of the 12 cases are summarized as follows, which can be seen in Table 3. Moreover, some fixed parameters that were constant in all of the cases are listed in Table 4.  Table 4. Fixed parameters of 12 simulation cases considered in this paper.

Elevation Angle/
With scissors angle difference = 0 • , noise SPL reaches relatively low values when left and right compass angles satisfy the relationship of: With scissors angle difference = 60 • , noise SPL reaches relatively low values when left and right compass angles satisfy the relationship of: With scissors angle difference = 30° or 90°, noise SPL reaches relatively low values when left and right compass angles satisfy the relationship of:  The highest SPL and range of relatively low SPL values at an azimuth angle = 0 • in helicopter mode are listed in Table 5, indicating that the noise SPL at an azimuth angle = 0 • (forehead) can be reduced by 7.4~10.4 dB after CSM phase modulation.  Figure 11 shows the noise SPL distributions on carpet diagrams with scissors angle difference = 0 • , 30 • , 60 • and 90 • at an azimuth angle = 90 • (side) in helicopter mode. It can be observed from Figure 11 Figure 11 shows the noise SPL distributions on carpet diagrams with scissors angle difference = 0°, 30°, 60° and 90° at an azimuth angle = 90° (side) in helicopter mode. It can be observed from Figure 11 that the distribution images of noise SPL values on carpet diagrams have similar shapes but different positions, with the same scissors angle difference. The relatively low values (blue color bars) also form obviously visible lines when scissors angle difference = 0° or 60° and are separated into points when scissors angle difference = 30° or 90°. However, the positions are different compared with Figure 10. When the scissors angle difference = 30° or 90°, noise SPL reaches relatively low values when left and right compass angles satisfy the relationship of: When the scissors angle difference = 0°, noise SPL reaches relatively low values when left and right compass angles satisfy the relationship of: When the scissors angle difference = 60°, noise SPL reaches relatively low values when left and right compass angles satisfy the relationship of: The maximum and minimum SPL values at an azimuth angle = 90° in helicopter mode are listed in Table 6, indicating that the noise SPL at an azimuth angle = 90° (side) can be reduced by 9.6 dB at most after CSM phase modulation.
When the scissors angle difference = 0 • , noise SPL reaches relatively low values when left and right compass angles satisfy the relationship of: When the scissors angle difference = 60 • , noise SPL reaches relatively low values when left and right compass angles satisfy the relationship of: The maximum and minimum SPL values at an azimuth angle = 90 • in helicopter mode are listed in Table 6, indicating that the noise SPL at an azimuth angle = 90 • (side) can be reduced by 9.6 dB at most after CSM phase modulation.  Figure 12 shows the noise SPL distributions on carpet diagrams with a scissors angle difference = 0 • , 30 • , 60 • and 90 • at an azimuth angle = 180 • (rear) in helicopter mode. Compared with Figure 10, it can be discovered that the distributions of noise SPL values have almost the same shape and position at an azimuth = 0 • and 180, and left and right compass angles also satisfy the relationship of Equations (39) and (40) so that noise SPL can reach relatively low values. Table 7 shows the maximum and minimum SPL values at an azimuth angle = 90 • in helicopter mode. Compared with Table 6, the noise SPL values at azimuth angles = 0 • and 180 • are also close to each other.  Figure 12 shows the noise SPL distributions on carpet diagrams with a scissors angle difference = 0°, 30°, 60° and 90° at an azimuth angle = 180° (rear) in helicopter mode. Compared with Figure 10, it can be discovered that the distributions of noise SPL values have almost the same shape and position at an azimuth = 0° and 180, and left and right compass angles also satisfy the relationship of Equations (39) and (40) so that noise SPL can reach relatively low values. Table 7 shows the maximum and minimum SPL values at an azimuth angle = 90° in helicopter mode. Compared with Table 6, the noise SPL values at azimuth angles = 0° and 180° are also close to each other.

Carpet Diagrams of Noise SPL Values in Fixed-Wing Mode
The noise SPL distributions on carpet diagrams with scissors angle difference = 0 • , 30 • , 60 • and 90 • at azimuth angles = 0 • (forehead), 90 • (side) and 180 • (rear) in fixed-wing mode are shown in Figures 13-15. According to the comparisons of noise SPL carpet diagrams in helicopter mode and fixed-wing mode, it can be indicated that the distribution rules of SPL values on carpet diagrams are similar at the same scissors angle difference, but the SPL values in helicopter mode and fixed-wing mode are quite different. The noise SPL values for all azimuth angles in fixed-wing mode are listed in Tables 8-10, indicating that the noise produced by coaxial tilt-rotor aircraft in fixed-wing mode is more significant than that in helicopter mode. Moreover, it can be discovered from Figures 13-15 that the noise SPL values at an azimuth angle = 0 • are obviously higher than those at an azimuth angle = 90 • and 180 • . The reason for this is that when the coaxial-tilt aircraft maintains high-speed forward flight, the air compression at the forehead is more significant than that at the side and rear, therefore generating higher noise.

Carpet Diagrams of Noise SPL Values in Fixed-Wing Mode
The noise SPL distributions on carpet diagrams with scissors angle difference = 0°, 30°, 60° and 90° at azimuth angles = 0° (forehead), 90° (side) and 180° (rear) in fixed-wing mode are shown in Figures 13-15. According to the comparisons of noise SPL carpet diagrams in helicopter mode and fixed-wing mode, it can be indicated that the distribution rules of SPL values on carpet diagrams are similar at the same scissors angle difference, but the SPL values in helicopter mode and fixed-wing mode are quite different. The noise SPL values for all azimuth angles in fixed-wing mode are listed in Tables 8-10, indicating that the noise produced by coaxial tilt-rotor aircraft in fixed-wing mode is more significant than that in helicopter mode. Moreover, it can be discovered from Figures 13-15 that the noise SPL values at an azimuth angle = 0° are obviously higher than those at an azimuth angle = 90° and 180°. The reason for this is that when the coaxial-tilt aircraft maintains high-speed forward flight, the air compression at the forehead is more significant than that at the side and rear, therefore generating higher noise.

Carpet Diagrams of Average RCS in Helicopter Mode
The calculated dynamic RCS of a coaxial tilt-rotor aircraft consists of the RCS values at all of the time points in a period of time, and the RCS values change as time flows. Average RCS in one period of rotor rotation is calculated in order to give a measurement of the RCS level of coaxial tilt-rotor aircraft at a specific azimuth angle and flight status, and carpet diagrams of average RCS are put forward in this paper.

Carpet Diagrams of Average RCS in Helicopter Mode
The calculated dynamic RCS of a coaxial tilt-rotor aircraft consists of the RCS values at all of the time points in a period of time, and the RCS values change as time flows. Average RCS in one period of rotor rotation is calculated in order to give a measurement of the RCS level of coaxial tilt-rotor aircraft at a specific azimuth angle and flight status, and carpet diagrams of average RCS are put forward in this paper.

Average RCS Values at an Azimuth Angle = 0 • (Forehead)
The carpet diagrams of average RCS in helicopter mode at an azimuth angle = 0 • can be seen in Figure 16. It can be observed that the scissors angle difference can influence the The carpet diagrams of average RCS in helicopter mode at an azimuth angle = 0° can be seen in Figure 16. It can be observed that the scissors angle difference can influence the distribution as well as the size of the RCS values on the carpet diagrams. Similar to the distributions of SPL values, the relatively low RCS values (dark blue color bars) also form obviously visible lines when scissors angle difference = 0° and 60° and also are relatively separated into points when scissors angle difference = 30° and 90°. Moreover, the average RCS values in helicopter mode at an azimuth angle = 0° also reach relatively low values when left and right compass angles satisfy the relationship of Equation (38)  The maximum and minimum average RCS values at an azimuth angle = 0° in helicopter mode is listed in Table 11, indicating that the noise SPL at an azimuth angle = 0° (forehead) can be reduced by 1.5 dBsm at most after CSM phase modulation. The maximum and minimum average RCS values at an azimuth angle = 0 • in helicopter mode is listed in Table 11, indicating that the noise SPL at an azimuth angle = 0 • (forehead) can be reduced by 1.5 dBsm at most after CSM phase modulation. It can be noticed from Figure 17 and Table 12 that the average RCS values in helicopter mode at an azimuth angle = 90 • are always close to 25.6 dBsm (green and light blue color bars). The maximum average RCS value reaches 25.6259 dBsm, while the minimum average RCS value reaches 25.5849 dBsm, with a variation range of only 0.041 dBsm. In addition, the average RCS values observed from the sides are much higher than those observed from the head and from the rear. However, the average RCS observed from the side remains almost the same value despite the variations in compass angles and scissors angle differences. The reasons can be explained by the radar scattering effect of the fuselage, which appears to be more significant in the side directions, leading to higher RCS values. Moreover, the existence of the fuselage brings shadow effects to the coaxial rotors, reducing the influence of compass angles and scissors angle difference on dynamic RCS. Compared with Figure 11, it can be inferred that the optimization of compass angles and scissors angle differences mainly contributes to the reduction of noise. It can be noticed from Figure 17 and Table 12 that the average RCS values in helicopter mode at an azimuth angle = 90° are always close to 25.6 dBsm (green and light blue color bars). The maximum average RCS value reaches 25.6259 dBsm, while the minimum average RCS value reaches 25.5849 dBsm, with a variation range of only 0.041 dBsm. In addition, the average RCS values observed from the sides are much higher than those observed from the head and from the rear. However, the average RCS observed from the side remains almost the same value despite the variations in compass angles and scissors angle differences. The reasons can be explained by the radar scattering effect of the fuselage, which appears to be more significant in the side directions, leading to higher RCS values. Moreover, the existence of the fuselage brings shadow effects to the coaxial rotors, reducing the influence of compass angles and scissors angle difference on dynamic RCS. Compared with Figure 11, it can be inferred that the optimization of compass angles and scissors angle differences mainly contributes to the reduction of noise.   Figure 18 shows the carpet diagrams of average RCS in helicopter mode at an azimuth angle = 180 • . It can be seen that the relatively high RCS values (yellow and green color bars) form clear lines while the relatively low RCS values (blue color bars) are rather separated. However, the vague lines formed by the relatively low RCS values can still be roughly seen in Figure 18a Figure 18 shows the carpet diagrams of average RCS in helicopter mode at an azimuth angle = 180°. It can be seen that the relatively high RCS values (yellow and green color bars) form clear lines while the relatively low RCS values (blue color bars) are rather separated. However, the vague lines formed by the relatively low RCS values can still be roughly seen in Figure 18a,c, with scissors angle difference = 0° and 60°. The vague lines indicate that average RCS can also reach relatively low values when left and right compass angles satisfy the relationship of Equation (39) at scissors angle difference = 0°, and Equation (40) at scissors angle difference = 60°. The average RCS values are about 4.1~4.2 dBsm, which are also relatively low values in the carpet diagrams. As for the situation when the scissors angle difference = 30° and 90°, the noise SPL can reach relatively low values when left and right compass angles satisfy the relationship of Equation (38). However, the average RCS reaches approximately 4.7 dBsm, represented by the light green bars (not relatively low values) in the carpet diagrams. Therefore, the noise SPL and the average RCS at an azimuth angle = 180° (rear) can reach relatively low values at the same time when the scissors angle difference = 0° and 60° but cannot when the scissors angle difference = 0° and 60°. The maximum and minimum average RCS values in helicopter mode at an azimuth angle = 180° are listed in Table 13. Compared with Tables 11 and 12, the minimum average RCS in helicopter mode at an azimuth angle = 180° appears to be lower than that at an azimuth angle = 0° and 90°. The reason can be explained by the fact that the radar scattering effect of the tail is weaker than that of the cabin at the forehead and much weaker than the side face of the fuselage. Table 13. Highest and lowest average RCS values at an azimuth angle = 180° in helicopter mode.  Table 13. Compared with Tables 11 and 12, the minimum average RCS in helicopter mode at an azimuth angle = 180 • appears to be lower than that at an azimuth angle = 0 • and 90 • . The reason can be explained by the fact that the radar scattering effect of the tail is weaker than that of the cabin at the forehead and much weaker than the side face of the fuselage. The carpet diagrams of average RCS in fixed-wing mode at an azimuth angle = 0 • can be seen in Figure 19. Analogously, the relatively low RCS values (dark blue color bars) also form obviously visible lines when the scissors angle difference = 0 • and 60 • , and also relatively separate points when the scissors angle difference = 30 • and 90 • . However, the directions of the lines in Figure 19a,c is vertical to the directions of the lines in Figure 16a,c. Therefore, in fixed-wing mode at an azimuth angle = 0 • , the average RCS can achieve relatively low values when left and right compass angles satisfy the relationship of: When the scissors angle difference = 60 • , left and right compass angles should satisfy the relationship of: The points illustrated in Equation (41)  The maximum and minimum average RCS values in fixed-wing mode at an azimuth angle = 0 • are listed in Table 14. By CSM phase modulation, the average RCS in fixed-wing mode at an azimuth angle = 0 • can be reduced by 1.8 dBsm at most. Combined with Figure 13, it can be indicated that the situations of scissors angle difference = 0 • and 60 • might be better for RCS reduction as well as noise reduction.  Figure 20 shows the carpet diagrams of average RCS in fixed-wing mode at an azimuth angle = 90 • and the maximum and minimum values are listed in Table 15. Similar to the situation in helicopter mode, the average RCS values in fixed-wing mode at an azimuth angle = 90 • are always close to 29.5 dBsm, remaining almost the same. The maximum average RCS value reaches 29.5128 dBsm, while the minimum average RCS value reaches 29.4957 dBsm, with a variation range of only 0.0171 dBsm. Moreover, the average RCS values in fixed-wing mode at an azimuth angle = 90 • are generally higher than those in helicopter mode. The reason can be explained by the fact that the scattering effects of cabins are more significant under the fixed-wing mode because the projection areas of cabins are larger.   Table 15. Similar to the situation in helicopter mode, the average RCS values in fixed-wing mode at an azimuth angle = 90° are always close to 29.5 dBsm, remaining almost the same. The maximum average RCS value reaches 29.5128 dBsm, while the minimum average RCS value reaches 29.4957 dBsm, with a variation range of only 0.0171 dBsm. Moreover, the average RCS values in fixed-wing mode at an azimuth angle = 90° are generally higher than those in helicopter mode. The reason can be explained by the fact that the scattering effects of cabins are more significant under the fixed-wing mode because the projection areas of cabins are larger. The maximum and minimum average RCS values in fixed-wing mode at an azimuth angle = 180 • are listed in Table 16. By CSM phase modulation, the average RCS in fixedwing mode at an azimuth angle = 180 • can be reduced by 1.8 dBsm at most. sors angle difference = 0° and 60°, when left and right compass angles satisfy the relationship of Equations (39) and (40) in order to obtain relatively low noise SPL, the lines formed by relatively high average RCS values can also be avoided, making contributions to the reduction of RCS at an azimuth angle = 180°. As for the situations where the scissors angle difference = 30° and 90°, the average RCS reaches 3 dBsm when compass angles satisfy Equation (38).

Optimal CSM Phase Parameters
In order to achieve the goal that both noise SPL and average RCS reach relatively low values, proper CSM phase parameters, including compass angles (left and right) and scissors angle difference, should be selected. Considering that coaxial tilt-rotor aircraft may switch from helicopter mode to fixed-wing mode or from fixed-wing mode to helicopter mode, optimal CSM phase parameters should be determined for the reduction of noise and RCS in both of the two modes.  The distribution rules of dynamic RCS at an azimuth angle = 180 • are rather vague, according to Figures 18 and 21. Considering that the dynamic RCS values and fixed-wing noise SPL values at an azimuth angle = 180 • are obviously lower than those at an azimuth angle = 0 • and 90 • , the reduction of noise SPL at an azimuth angle = 180 • in helicopter mode should be given the most attention. Therefore: At scissors angle difference = 0 • , noise SPL reaches relatively low values when left and right compass angles satisfy the relationship of: At scissors angle difference = 30 • , noise SPL reaches relatively low values when left and right compass angles satisfy the relationship of: At scissors angle difference = 60 • , noise SPL reaches relatively low values when left and right compass angles satisfy the relationship of: At scissors angle difference = 90 • , noise SPL reaches relatively low values when left and right compass angles satisfy the relationship of: With the CSM phase parameters of Equations (51)-(54), the noise SPL and average RCS values are listed in Table 19. At scissors angle difference = 0 • , the optimal CSM phase parameters should be: At scissors angle difference = 60 • , the optimal CSM phase parameters should be: According to the periodicity and symmetry of coaxial rotors, the two CSM phase parameter combinations represent the same condition for coaxial tilt-rotor aircraft, as is shown in Figure 22. The noise SPL and average RCS values are listed in Table 20. The CSM phase parameter combinations of Equations (55) and (56)   Azimuth Figure 22. View of a coaxial tilt-rotor aircraft with PPC1.  (48) and (50). This illustrates that we have to make a choice between the forehead/rear and side when the scissors angle difference = 30 • or 90 • .
With priority given to noise and RCS reduction of the forehead/rear at a scissors angle difference = 30 • , the optimal CSM phase parameters should be: With priority given to noise and RCS reduction of the side at a scissors angle difference = 30 • , the optimal CSM phase parameters should be: With priority given to noise and RCS reduction of the forehead/rear at a scissors angle difference = 90 • , the optimal CSM phase parameters should be: With priority given to noise and RCS reduction of the side at a scissors angle difference = 90 • , the optimal CSM phase parameters should be: Ignoring the influence of the distance between the lower and upper rotors of a coaxial rotor, the CSM phase parameter combinations Equations (57) and (59) represent the same status of aircraft, as is shown in Figure 23. The noise SPL and average RCS values are listed in Table 21. Similarly, CSM phase parameter combinations Equations (58) and (60) represent another status of aircraft shown in Figure 24, and Table 22          According to the comparisons of Tables 20-22, the average RCS values of the three PPCs have little difference with each other, but the noise SPL values obviously vary from each other. The design with PPC2 has a lower noise SPL at an azimuth angle = 0 • (forehead) and 180 • (Rear), while the design with PPC3 has a lower noise SPL at an azimuth angle = 90 • (side). Table 23 shows the reduction values (compared with maximum values) of noise SPL and average RCS of PPC1, PPC2 and PPC3. noise reduction at the forehead and rear, and the stealth design with PPC3 can contribute most to noise reduction at the side.

Conclusions
In this paper, an approach to the prediction of both the sound pressure level (SPL) of noise and the dynamic RCS of coaxial-tilt aircraft is carried out based on the theories of FW-H equation. A dynamic RCS and noise prediction and reduction method for coaxial tilt-rotor aircraft based on phase modulation is established with the generated rotation matrix (GRM), compass-scissors model (CSM) and carpet diagrams put forward.
Compared with the existing work, the establishment of GRM gives a universal formula for the time-domain grid coordinate transformation of all kinds of rotation parts with arbitrary rotation centers and rotation axis directions. As a newly established model, CSM expresses the description of the orientation and the rotation phase status of coaxial rotors, making research on the phase characteristics of coaxial rotors possible. Carpet diagrams applied in this paper intuitively show the variations of noise SPL and dynamic RCS under different phase parameters (defined in CSM) and can be helpful to the stealth design of tilt-coaxial rotor aircraft using phase modulation of the coaxial rotors. According to the results, the following conclusions can be obtained: Moreover, the average RCS in helicopter mode is lower than that in fixed-wing mode at an azimuth angle = 90 • but higher at an azimuth angle = 0 • and 180 • . (5) The stealth design with PPC1 can contribute most to RCS reduction, the stealth design with PPC2 can contribute most to noise reduction at the forehead and rear, and the stealth design with PPC3 can contribute most to noise reduction at the side. Optimal design can be selected according to the requirement of stealth based on the threats.