Longitudinal-Plane Simultaneous Non-Interfering Approach Trajectory Design for Noise Minimization Gaurav Gopalan♦

Min Xue♦

Ella Atkins*

Fredric H. Schmitz£

Alfred E. Gessow Rotorcraft Center University of Maryland at College Park College Park, MD 20742

g Abstract Runway-independent aircraft (RIA) operating under simultaneous non-interfering (SNI) traffic procedures have been proposed to alleviate airspace congestion at crowded urban airports. This paper introduces a methodology for designing minimum-noise longitudinal SNI approach trajectories for rotorcraft. An analytical model for ground noise annoyance associated with out-of-plane Blade-Vortex Interaction (BVI) noise is introduced and its application as the cost function for SNI trajectory optimization is described. The noise model relies on a physicsbased semi-empirical expression developed to approximate the average annoyance levels associated with BVI noise on a representative ground plane. To guarantee strictly SNI trajectories, fixed-wing traffic corridors are treated as impenetrable obstacles modeled by their cross-sections in the longitudinal approach plane. Two optimization procedures are employed: a heuristic strategy that specifies trajectories using a small approach waypoint set and a globally-optimal cell-based algorithm. The feasibility and practicality of example minimumBVI noise solutions are discussed.

g(X) I MHT SPL SEL T ∆t V & V

W x z

αTPP µ γ λ

= Acceleration due to gravity (32.2 ft/sec2) = Non-linear constraint function = Coefficients/Indices used for curve-fits = Hover-tip Mach number, ΩR/ao = Sound pressure level = Sound exposure level = Main rotor revolution period = Elemental time-step = Flight velocity = Acceleration parallel to flight path = Helicopter weight =Horizontal coordinate along the trajectory = Height of the helicopter above the ground = Main rotor tip-path-plane angle (positive nose up) = Advance ratio, V/ΩR = Flight path angle (negative in descent) = Average rotor inflow ratio (positive for downwash)

Subscripts/Superscripts av = spatial averaging dB = Expressed in decibel final = End-point along the trajectory Ground = Associated with the ground plane i =Temporal index or time-step associated with the trajectory initial = Starting point for the trajectory n = Spatial index over the ground plane o = Peak level or reference value

Nomenclature = Ground plane total area Ao ∆A = Elemental area on the ground plane C = Coefficients/Indices used for curve-fits = Thrust coefficient CT = Effective drag force acting on the Deff helicopter E = Total energy, kinetic plus potential, of the helicopter F(X) = Objective function

Introduction The world’s need to travel continues to put tremendous growth pressure on the commercial airline system. Because few new airports are being planned, many airports operate at or near capacity with many others approaching a similar state. Runway-independent aircraft (RIA) defined as either VTOL (vertical takeoff and landing) or

♦ Graduate Research Assistant * Assistant Professor £ Martin Professor of Rotorcraft Acoustics Presented at the 59th AHS International Forum and Technology Display, Phoenix, Arizona, May 2003. © 2003 by the American Helicopter Society, International. All rights reserved.

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eSTOL (extremely short takeoff and landing) have been proposed as an alternative for short to medium range (<400 nm) flights. Because of their unique takeoff and landing capabilities, RIA will be based at the same airports but at locations that will not interfere with the fixed wing traffic flow. This non-interfering operation allows the shorter fixed-wing traffic slots to be filled by longer-range, higher-passenger revenue flights, thus substantially increasing the overall capacity of the airport.

navigation aids and standard instrument procedures, with airline and air traffic control preferences guiding selection when multiple options are available. Betts2 presents a thorough review of direct and indirect numerical optimization methods. Seywald3 et al and 4 Schultz discuss trajectory optimization for aircraft flying in the vertical plane using a point mass performance model. The simplification to a two-dimensional problem is common for trajectory synthesis because changes in heading are negligible for time and fuel minimization purposes. Hagelauer5 proposes an approach to flight path optimization based on dynamic programming, while Slattery and Zhao6 study trajectory synthesis for air traffic management to enable controllers to better guarantee safety and increase efficiency via reduced spacing.

Introduction of new RIA traffic patterns in crowded terminal airspace has the potential to greatly increase air traffic controller workload, creating new conflict-related delays rather than alleviating congestion. Simultaneous NonInterfering (SNI) approach and departure procedures have been proposed to minimize air traffic control overhead and maximize overall throughput1. SNI paths do not intersect existing traffic corridors, so arrivals and departures can be sequenced independent of fixed-wing traffic. By definition, SNI trajectories occupy previously unused airspace thus may traverse areas previously undisturbed by fixed-wing traffic.

Low-dimensional parameter searches are first used to derive simple near-optimal solutions without en-route obstacles to give rationality to the optimization process. This approach uses gradient descent to find low noise, flyable trajectories composed of a small set of waypointbased segments. Next, a cell decomposition approach to global optimization7 with modified quad-tree cell construction was adapted to the SNI airspace optimization task due to its ability to handle arbitrary obstacles and complex cost functions. Once the model has been generated, a uniform-cost search8 is conducted to identify the minimum-noise trajectory through the space of cells. A number of case studies are presented to characterize noise-minimum trajectories for varied boundary conditions, dynamic constraints, and airspace obstacle placements.

Many of these airports are also constrained by community noise issues. Although the higher bypass ratios of newer airplanes have decreased the radiated noise per pound of gross weight, the increasing number of airplane operations continues to make noise a key issue of airport growth. If VTOL aircraft are to replace fixedwing aircraft over these shorter route segments, they must not increase the noise radiated to the communities surrounding the airport.

Helicopter Blade-Vortex Interaction (BVI) Noise Trends Although there are many sources of noise that are radiated from a conventional single rotor helicopter, when it occurs, helicopter impulsive noise is the most dominant and objectionable. It's characteristic blade "slapping" sound stands out from other noise sources and is often the stated target of community action against helicopter operations. One cause of helicopter impulsive noise arises when the helicopter's main rotor shed wake operates close to the rotor's tip-path-plane, causing rapid changes in the local angle of attack on the main rotor. These rapid changes in the induced velocity flow field at the rotor blades cause impulsive air-loads, which in turn, push on the air causing impulsive sources of sound, which

The goal of this research is to develop procedures that can automatically generate SNI final approach trajectories that minimize ground noise. A single rotor helicopter is chosen as the representative VTOL aircraft that is known to generate BVI noise during descent and deceleration to landing. The SNI final approach trajectory optimization is defined as a two-point boundary value problem in the longitudinal plane with fixed-wing airspace corridors modeled as impenetrable obstacles. Average radiated BVI noise over a ground plane is adopted as the sole measure of cost. Traditional aircraft trajectory synthesis tools have focused on minimizing fuel or flight time subject to vehicle dynamic and air traffic control constraints. Waypoints are based on ground

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result have been partially validated in several model-scale and full scale experiments on both two- and four-bladed rotors. The BVI noise levels shown in this paper are relative noise levels and are based upon a two-bladed AH-1 series sized helicopter. Thus, the rather large variations in noise levels as a function of flight conditions shown in this paper must be interpreted as only BVI noise reductions – not the total noise reduction possible for this class of helicopters. If BVI noise were reduced to very low levels, it is quite possible that other rotorcraft noise sources may set the lower levels of noise radiation.

has been labeled as Blade-Vortex Interaction (BVI) noise. A sketch of a side view of a helicopter with its shed wake is shown in Fig. 1-a for a helicopter in normal level flight. BVI noise is not seen as a problem in this nominal cruising condition because the shed wake is underneath the rotor for all potential BVI. However, when the helicopter descends or decelerates in preparation for a landing, the rotor's shed wake can be forced to operate in or very close to the rotor blades, causing strong BVI noise (Fig. 1-b). Further increases in descent/deceleration may cause the rotor wake to operate above the tip-path-plane of the helicopter (Fig. 1-c), causing reductions from the peak BVI noise levels9.

For simplicity, it has been assumed that BVI noise is only governed by rapid changes in lift on the main rotor. Because changes on in-plane drag have been neglected, only out-of-plane noise radiation with the helicopter operating in the longitudinal (X-Z) plane, is included. In steadystate flight, BVI noise is known to be a function of four main-rotor non-dimensional variables; advance ratio, µ, main rotor, tip-path-plane angle, αTPP, hover tip Mach number, MH, and thrust coefficient, CT. It is possible to calculate the BVI sound pressure levels over a one rotor revolution period on a sphere fixed to the stationary medium and centered about the mid-period location of the radiation sphere surrounding the helicopter as a function of these non-dimensional variables. When hover tip Mach number and thrust coefficient are assumed constant, the BVI noise on the spheres becomes a function of rotor advance ratio and tip-path-plane angle. A procedure known as "Q-SAM" (Quasi-Static Acoustic Mapping) has been used to extend these results for slowly maneuvering flight10. To first order, the quasi-static tip-path plane angle of the main rotor, during slowly accelerating flight is estimated as:

Figure1: Effect of flight path angle on inflow through the rotor disk: a) level flight, b) nominal descent and c) steep descent approach.

α TPP = −

There have been many attempts to accurately model the BVI noise generation/radiation process with varying degrees of success. Most attempts rely on reliable estimates of the details of the shed wake, the unsteady aerodynamic response of the rotor to the disturbance produced by the shed wake and the acoustics characteristics of the interaction process. This paper focuses on the noise level trends of BVI10, and as such, uses a model that relies on a semi-empirical "Beddoes" wake11. The noise radiation characteristics that

& D eff V −γ− W g

(1)

Using Q-SAM, it is possible to specify a helicopter flight path (acceleration and flight path angle) time history and calculate the effective radiation spheres along the trajectory, as well as the resulting SPL time history and the SEL at any “far-field” observer location. Typically, the results are presented as an SEL contour plot over a ground plane. The average value of SEL over this plane computed on an energy basis is referred to as SELav. For the trajectory optimization

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problem, it is proposed to use the SELav as a cost function, for determining an acceptable trajectory. SELav was defined in Reference 10 as:

SPLGround = SPL av,dB



− 20log10 1 + I 2µ 2  α  

F = SEL av = Ground  Trajectory Plane  Elements

∑

10log10

∑

   

n

10

To

  

where,

Ao

I ,  2,1  I2,1 =   I2,2 , 

In the above expression “n” refers to a spatial location on the observer plane and “i” refers to an element along the trajectory. By reversing the order of summation over the spatial and temporal dimensions (Appendix A.1), the area-average SEL over the ground plane can be approximated as the time-summation of the area-average SPL over the ground plane associated with each element along the flight trajectory:

∑

10

SPLGround (z i , µ i , α TPP, i ) 10 av, dB

∆t i To

i

where: SPLGround (zo , µ, α TPP ) = av,dB

10log10

∑ n

SPL dB, n

10

∆An ∆t n Ao

 if  αTPP < α  TPP,0,eff    if  αTPP > α  TPP,0,eff  

This model is based on the physics of individual BVI and extended to the overall trend of BVI noise radiation as a function of advance ratio and tip-path plane angle (see Appendix A.2). Its application to ground noise trends is based on the previous observation that the variation of the average BVI sound pressure levels on the ground plane closely follows the average BVI sound pressure levels radiated by the helicopter over a radiation sphere. The resulting curve-fit is compared with analytical data in Figure 3 for an advance ratio of 0.165, which corresponds to a flight velocity of about 70 knots for the AH-1 helicopter. Further details are presented in Appendix B. Equation 5 correlates with trends corresponding to higher advance ratios better than lower advance ratios. The overall correlation is reasonable for the purpose of this study, over the entire range of advance ratios. Typically, at a fixed advance ratio, as the flight path angle is varied from zero degrees in level flight to steeper approach angles, the average radiated sound power associated with BVI noise increases to a maximum value and then begins to reduce again. This variation of the average BVI ground noise radiation with tip-path-plane angle corresponds to the wake effectively operating below the disk at small tip-path-plane angles and shallow flight path angles, cutting through or near the rotor disk at intermediate tip-path-plane angles, and finally being pushed above the rotor disk for steep descent flight conditions which correspond to higher tip-path-plane angles.

(3)

Ground Plane

(5)

∆t i,n  ∆A n

(2)

F ≈ 10log10

 TPP,0,eff

2    

- αTPP 



SPLdB,i,n 10

i

Trajectory Elements

o, dB

(4)

T

where, SPLdB,,n refers to the sound pressure level received at an observer, denoted by the index n. ∆An represents the area element associated with the observer location. ∆tn refers to the difference between the arrival times of the noise signal from the two end-points of the element. T is the time period associated with a trajectory element, typically one rotor revolution period. Ao refers to the area of the ground observer plane. The pre-computed trends of the average SPL on the ground plane, shown in Fig. 2 for different advance ratios in the range 0.12 and 0.21, are curve-fit to a semi-empirical model that is based on the physics of the BVI noise generation process (Appendix A.2). For a fixed height z above the ground, this trend is expressed as (Appendix A.2):

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Figure 2: SPLav over the ground plane for steady flight conditions as a function of tip-path plane angle for advance ratios 0.12, 0.143 0.165, 0.188 and 0.21.

Figure 4: The variation of the mean-trend for the radiated main-rotor BVI sound power averaged over a representative ground plane plotted as a function of height above the ground plane center.

The effect of atmospheric absorption on the noise levels as a function of propagation distance is, in general, a function of the power spectrum or frequency content of the noise signature, and therefore of the flight condition. For the data set used however, it is observed that the variation with z, of average A-weighted noise levels for any flight condition is independent of flight condition, within an error bound of about 1 dB. Therefore, in the current model, the variation of average radiated noise levels as a function of height above the ground plane is assumed to be independent of flight conditions. The dependence of average radiated noise on the ground on the height z of the helicopter above the ground is estimated as below (Fig. 4): SPLGround (z,µ , αTPP ) = SPLGround (zo , µ , αTPP ) av,dB

av,dB

Optimization Problem Statement: Based on the objective function F, described above in Eq. 2, the task of finding an optimal longitudinal approach trajectory, in terms of the minimum associated BVI noise annoyance, can be expressed mathematically as follows: Find the function z(t), µ(t) and αTPP(t) that, subject to a set of specified initial and final conditions, minimize the function: F = 10log10

t final ∫

10

SPLGround (z, µ, α TPP ) 10 av, dB

t initial

(6)

= 10log10

+ ∆SPL(z, zo )

t final

dt To

(7)

∫ f(z, µ, α TPP )dt

t initial

where z& = Vsin γ

µ = V ΩR

& D eff V −γ− . W g It is assumed that the main rotor tip-speed is held approximately constant during nominal approach trajectories and that the effective drag of the helicopter is primarily a function of the flight velocity. Using the relations above, the optimization problem statement can now be posed as:

α TPP = −

Figure 3: Curve-fit for the average radiated BVI sound power on the ground plane, as a function of main-rotor tip-path plane angle, for advance ratio 0.165.

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& (t ) and γ(t) that minimize Find the functions V the function:

F = 10log10

t final

f(z (t ), V (t ), γ (t ) + V (t ) )dt g

γ≈

& (x ) ∂ V 2 (x) 1 ∂E = γ (x ) + V and = γ (x ) + g ∂x g ∂x 2

&

∫

t initial

(8)

2 E = gz(x ) + V (x)

where, t

z(t ) = zinitial +

∫

Vsin γ dt

V (t ) = Vinitial +

t ∫

& dt V

t initial

The acceleration along the flight path and the flight path angle along the trajectory are treated as the “controls” of the problem. The focus of the current project is on nominal descent approach conditions. Therefore, certain key aspects of the performance characteristics of the helicopter can be expressed in terms of bounds on the behavior of functions γ and V and their derivatives, and an idealized approximation to an approach trajectory profile is developed. It is first assumed that the flight path angle is restricted to climbs and descents no steeper than 9° at all flight speeds. A bound is also imposed on the maximum acceleration or deceleration parallel to the flight path. Based on passenger comfort this value us selected as 0.05g. The flight velocity V is restricted to lie between 40 knots and 100 knots, a typical range for nominal approach procedures. The height z of the helicopter is also bound between 50 feet and 2000 feet.

∫

x initial

≡ 10log10

x final ∫

x initial

zi,i −1(x ) ≈ zi −1 − γ i,i −1[x − x i −1 ] Vi,2i −1 (x )

& dx f(z (x ), V (x ), γ(x ) + V (x ) ) g - V(x)

dx f(z (x ), V (x ), 1 ∂E ) g ∂x - V(x)

.

During a nominal approach to a landing, a helicopter pilot typically executes a small number of constant glide-slope segments before the final stages of flare and touch down. The very last stages of descent are usually close enough to the heliport to not count in terms of their acoustic impact on noise sensitive areas. Typically these longitudinal approach trajectories consist of a series of constant flight path angle segments with constant acceleration along the flight path. Therefore, for trajectories under consideration, each constituent segment is characterized by a constant flight path angle, a uniform acceleration along the flight path and a segment length. The functions z and V2/2, which represent the potential and kinetic energies of the helicopter respectively, are further idealized to be piecewiselinear continuous functions along the trajectory:

The small angle assumption made for the flight path angle and the consideration of only longitudinal trajectories impose a condition of strict monotonocity on x(t). The objective function can therefore be cast as an integral over x rather than time. The initial and final x locations become the bounds of x. The objective is to find functions z(x) and V(x) that minimize the function F subject to the boundary conditions, and the problem constraints. For travel in the reverse x direction: x final

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The term E(x) represents the sum of the kinetic and potential energies associated with the helicopter. The flight path angle is therefore equivalent to the gradient of the potential energy of the helicopter along the trajectory, while the acceleration along the flight path is equivalent to the gradient of kinetic energy. The objective function therefore depends only on z(x) and V(x) and their derivatives as a function of x. While z(x) completely specifies the trajectory geometry, V(x) imposes a dynamic character to the trajectory in terms of a velocity profile. It should be noted that the objective function, based on the BVI noise radiation characteristics of the helicopter, couple the choice of the functions z(x) and V(x), preempting the possibility of selecting them independently.

tinitial

F = 10log10

∂z ∂ V 2 (x) & ≈ and V ∂x ∂x 2

2

≈

Vi2−1

2

& −V

i,i −1

[x − x i−1 ]

The sequence {xi zi Vi} that constitute the end points of these piecewise linear functions are called waypoints or node points of the trajectory. Such a trajectory is uniquely defined by the sequence {xi , zi , Vi}, representing the values of x, z and V at each waypoint. The boundary

(9)

where,

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conditions for such a trajectory with n segments, and n-1 waypoints between the specified boundary points, are expressed as {xo , zo , Vo} and {xn , zn , Vn}. The objective function for this discretized system, can be expressed as: F = 10log10

n xi

∑∫ i =1

& f(z i,i-1 (x ), Vi,i -1 (x ), γi,i -1 + V i, i -1 )

x i -1

= 10log10

n xi

∑∫ i =1

f(z i,i -1 (x ), Vi,i -1 (x ),

x i -1

along with the two boundary points, divide the trajectory into n segments. For a specified set of boundary conditions, the waypoint locations and the associated flight velocity completely specify the entire trajectory. Therefore, for a trajectory composed of “n-1” waypoints:

dx - Vi,i-1 (x )

X = [x 1 , z1 , v1 ,...x i , z i , v i ,...x n -1 , z n -1 , v n -1 ] :

1 E i − E i -1 dx ) g x i -1 − x i - Vi,i -1 (x )

Design Vector X initial = [x 0 , z 0 , v 0 ]

(10)

X final = [x n , z n , v n ] :

where,

Boundary Conditions (Initial and Final Approach Fixes)

γi,i −1 ≈

& V i, i −1 ≈

zi - zi −1 x i −1 − x i

V2 i

F(X) : Objective Function, Equation 10

- V2 i -1

2(x i −1 − x i )

V2 Ei = gz i + i

X LB ≤ X ≤ X UB : Lower and Upper Bounds on X & & & & g(X) = [V i,i -1 - Vmax ,-Vi,i -1 + Vmin ,

2

γi,i -1 - γ max ,- γi,i-1 + γ min ] ≤ 0 :

This idealization of the trajectory may introduce discontinuities in dz/dx and dv/dx at the node points. Transient maneuvers like changes in flight path angle and changes in the acceleration along the flight path are governed by the vehicle dynamics, performance and stability and control equations, and in turn affect the noise characteristics of the helicopter. Therefore constraints on the variation of acceleration and flight path angle along such a segmented trajectory should also be introduced to represent the physics of these transient unsteady phenomena. At the initial stage, these transient effects have not been modeled.

Non - linear constraints (Bounds on accelration and flight path angle)

Starting with a relatively small number of segments, an optimal solution is sought. The characteristic nature of BVI noise allows for nonunique locally optimal solutions under nominal approach conditions. By varying the initial value of the design vector Xo, potentially, several local minima can be found for a given set of boundary conditions. By introducing one new waypoint along any segment of the set of local minimum solutions for an n-segmented trajectory, possibly the segment with the highest contribution to the objective function, initial values for the design vector of an n+1-segmented trajectory are generated. This process is repeated till a trajectory with an acceptable value for the objective function is obtained. Depending on the nature of the objective and constraint functions as well as the bounds on the problem variables, arriving at the global minimum value of the objective function may potentially require a very large number of segments. But such a trajectory may be unrealistic in terms of its actual implementation by pilots and air-traffic controllers. It may be of interest to find the minimum noise trajectory for an approach consisting of a small number of segments, for a given set of conditions and

Two distinct computational methods are used for trajectory optimization in this paper – gradient descent to find the local optimum solution for a small-number of waypoints and a uniform-cost based search approach that relies on a celldecomposition of the longitudinal plane to find the global optimum solution. Results obtained from these methods are compared and the scope of each of these two methods is discussed in the context of the development of SNI approach trajectories that minimize noise on the ground. Trajectory Optimization Using Gradient Descent An n-segmented approach trajectory is assumed to consist of a series of “n-1” waypoints, which,

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constraints. In effect, this would yield the most acoustically efficient n-segmented trajectory.

form of the objective function, care must be taken when trying to arrive at the global minimum solution. Also it is of greater interest to arrive at a range of segment parameters that correspond to a low value of the objective function, rather than a single trajectory that corresponds to the minimum associated value of SELav.

Figure 5 shows some solution trajectories for a range of 50,000 feet and a constant flight velocity of 70 knots. The boundary values were chosen as x = 0 ft, z = 50 ft and x = 50,000 ft, z = 1000 ft. The z dimension was bound by 50 ft and 1500 ft. Flight path angles were constrained to lie between climbs and descents of 9° and accelerations were constrained to be 0. Starting at n=1, it is observed that simply joining the two boundary points actually yields a feasible solution, a shallow approach trajectory, which is one of the local minima of this problem. This trajectory is associated with an SELav of about 86 dB. By placing one waypoint along this trajectory, n is increased to 2, and the optimization procedure is performed. The solution converges to a very similar trajectory with a flight path angle close to -1°. By choosing other feasible initial conditions, two other minimum noise solutions are obtained. One minimum solution is a steep descent followed by a very shallow approach to the final point. This trajectory is associated with an SELav of about 84 dB. The other minimum solution, the global minimum for a two-segmented trajectory for the specified conditions and constraints, starts with a climb to 1500 feet followed by a steep descent to the final point. This trajectory corresponds to an SELav of about 78 dB. When climbs are restricted the minimum-noise solution consists of level flight followed by a steep descent. The two maximum-noise trajectories are also shown in Fig. 5. Tese trajectories consist of a 5 degree approach either preceded or followed by level flight. Such a trajectory corresponds to an SELav of about 92 dB.

Figure 5: Several low-noise solutions for a 2segmented trajectory with a range of 20000 feet, obtained using gradient descent. The maximum noise trajectories corresponding to 4.8 degree approaches are also shown for comparison.

Figure 6 shows similar trajectories for a horizontal range of 20000 feet. Because trajectories last for a shorter duration, the average annoyance levels associated with the corresponding trajectories are lower when compared to Fig. 5.

Figure 6: Several low-noise solutions for a 2segmented trajectory with a range of 20000 feet, obtained using gradient descent. The maximum noise trajectories corresponding to 4.8 degree approaches are also shown for comparison.

This relatively simple model is used to obtain low noise trajectories for a small number of segments. The choice of a suitable initial design vector to represent an initial choice of the trajectory is critical to the success of this method, especially when the design vector becomes large in size. This method is also extendable to three dimensions in space, at the cost of design vector size. It is to be noted that because of the approximations made in arriving at the analytical

Trajectory Optimization using CellDecomposition The BVI noise model is adopted as the cost function during final approach trajectory generation. The goal is to identify globally-

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optimal SNI trajectories given realistic constraints. An approximate cell decomposition method7 using modified quad-tree cell construction is used to define the longitudinalplane search space given the presence of airspace obstacles. The algorithm takes spatial boundaries (x, z), dynamic constraints (γ, V, V& ), and polygonal obstacle boundaries as inputs, and returns the set of cells to be searched for an optimal solution. Fundamental cell decomposition includes the following steps:

Figure 7: Examples of Approximate Decomposition and Modified Approximate Decomposition

1. Divide the geometric space into cells. Cells must be non-overlapping and of prespecified shape. (e.g. rectangular). 2. Construct the connectivity graph from adjacency relations. 3. Search the graph for channels between initial and goal configurations.

Cell Cell

Once the modified approximate cell decomposition map is created, this space must be explored to identify the optimal trajectory given boundary condition pair (xinitial , zinitial ,Vinitial) and (xfinal , zfinal ,Vfinal). Typical approaches include dynamic programming and A* search7, with an A* approach selected for this work due to its improved computational efficiency in the average case. A* explores nodes from initial to final (goal) state in best-first ordering based on an evaluation function f (n) . Let g(n) be the actual path cost from the start node (initial state) to current node n and h(n) be the estimated cost of the cheapest path from n to the goal. The overall evaluation function f (n) = g (n) + h (n), and it can be proven that A* yields an optimal result so long as h(n) is an admissible heuristic (i.e., never overestimates cost from current node to the final state). When h(n) = 0, A* search becomes uniform-cost search with evaluation function f(n) = g (n). All three search strategies provide optimal results, however with h(n)>0, A* search is “informed” thus typically more efficient in finding the optimal path. Given the complexity of the noise function, a decent admissible heuristic has not yet been identified thus the trajectory optimizer utilizes uniform-cost search with cost g(n) set to radiated ground noise summed over the path from initial state to node n.

A quad-tree data structure was used for cell decomposition. At every depth level of the quadtree, each cell is classified in one of three groups. The cell is defined as “empty” if and only if its interior does not intersect the obstacle region, “full” if and only if it is entirely contained in the obstacle region, otherwise as “mixed”. A cell is divided into four sub-cells of equivalent shape only when classified as “mixed”. This recursive procedure (algorithm Step 1) is repeated until no cells are mixed or the maximum specified depth level is reached. The left plot in Fig. 7 illustrates a typical decomposition. Because this method was originally developed for robotic vehicles with few dynamic constraints, modifications were required to fit the current problem. First, if there is no obstacle in the airspace or the final “empty” cells are too big, the numerical approximation will not have sufficient resolution. Thus, to find optimal solutions in all cases, both “empty” and “mixed” cells are divided as shown in the right plot of Fig. 7. Additionally, because a helicopter climbs/descends with flight path angle between ± 9 o , to allow for sufficient search-space resolution, the ratio between cell length and width is set to 100:1, which yields a path angle interval of about 0.6o . Next, each step in the original method will only search the neighbor nodes, defined as the eight or fewer cells geometrically adjacent to the current cell. Given rotorcraft dynamic constraints, e.g. if flight path angle ranges from –9° to 9°, adjacency is redefined as the nearest 32 nodes in the left (forward flight direction) neighbor column of each cell, giving with an interval of about 0.6°.

Noise-Optimal Longitudinal Approach Trajectories The cell decomposition approach was used to generate a number of globally-optimal approach trajectories that minimize the SELav over the ground plane. For all approach cases, the initial longitudinal position is x = 20,000 feet, z = 1000 feet and the final position is x = 1,500 feet and z = 50 feet. Solution trajectories corresponding to constant speed approaches are very similar to those obtained using the gradient-descent method

9

(Fig. 6). This indicates that the global optimal solution for constant speed approaches is actually composed of a small number of segments.

steep climbs and descents, with the climbs accompanied by acceleration and descents by deceleration. Again, each segment comprising the optimal trajectory is associated with high inflow and low BVI noise radiation. The average exposure level on the ground in general increases as the height above the ground decreases but this is seen to be a secondary factor to the noise radiation characteristics of the helicopter itself. A helicopter in an approach to landing typically starts at a low cruising velocity, say 100 knots, and as it approaches the ground further reduces its velocity in preparation for landing. For the remaining cases the helicopter starts with a velocity of 95 knots at an altitude of 1000 feet and ends at a velocity of 45 knots, 50 feet above the ground. The basic case allows the helicopter to descend and decelerate (Fig. 10) but not climb or accelerate. The corresponding optimal trajectory starts with a steep decelerating descent leveling out at about 100 feet with a velocity of about 50 knots. This level flight segment level is then followed by a shallow decelerating descent to the final velocity and location. As the flight velocity reduces, the induced velocity “down” through the rotor disk increases. A steep decelerating descent under such low velocity flight conditions would result in a low inflow through the rotor disk, increasing the level of BVI noise. Therefore the leveling out of the trajectory at low speeds is an attempt to keep the vortex structure sufficiently below the rotor disk to avoid high BVI noise radiation. If the helicopter is allowed to accelerate but not climb, the optimal solution trajectory remains unchanged because accelerated descents typically correspond to lower inflow values. If climbs and deceleration are allowed but not acceleration (Fig. 11), the basic trajectory profile of Fig. 10 is preceded by a constant speed climb segment which further reduces the value of the average SEL. If the helicopter is finally allowed to accelerate and decelerate in addition to climbing/descending (Fig. 12), the resulting optimal solution, a global minimum for the specified boundary conditions is a saw-toothed sequence of accelerated climbs and decelerating descents, a “bang-bang” solution. The final segment is a shallow low-speed decelerating approach, as in Figs. 10 and 11. This global optimum solution is compared with the “worst-noise” approach, a shallow descent approach that always maintains a relatively low value of inflow. All other solutions corresponding to this set of boundary conditions are constrained minima and as such are sub-optimal when the

Figure 8: Optimal approach trajectory for initial and final velocities of 70 knots – no climb allowed.

Figure 9: Optimal approach trajectory for initial and final velocities of 70 knot with climbs allowed.

For the same initial and final velocity boundary conditions, 70 knots, the helicopter is now allowed to accelerate and decelerate along the intermediate flight path such that the final velocity remains 70 knots. If the helicopter is not allowed to climb (Fig. 8), the resulting optimal segmented-trajectory is a sequence of steep decelerating descent segments and accelerating level flight segments. Both these flight conditions correspond to a large inflow through the rotor disk, “down” through the rotor disk for accelerating segments and “up” for decelerating segments. When the helicopter is now additionally allowed to climb (Fig. 9), the helicopter flight path is an alternating sequence of

10

artificial climb and acceleration constraints are relaxed.

solutions may be preferred to other entirely different local optima. This would also depend on where along the trajectory the obstacle is placed. One strategy to obstacle avoidance would be look at the different optimal solutions corresponding to a given set of boundary conditions, by the successive imposition of restrictions on the helicopter’s ability to climb or accelerate along the flight path, as in the previous section, and select the trajectory that is not obstructed by the imposed obstacles. If no such trajectory exists then determine neighboring sub-optimal solutions that may still be acceptable.

Figure 10: Optimal approach for initial velocity of 95 knots and final velocity of 45 knots – no climb.

Figure 12: Optimal Approach for initial velocity of

95 knots and final velocity of 45 knots -- allow deceleration, acceleration, climb and descent. Using the gradient descent approach, obstacle avoidance is achieved through the imposition of intersection constraints on xi and zi and by imposing a suitable potential function into the objective function that peaks in the interior of obstacles and slopes down in value towards the obstacle boundaries. This potential function is not allowed to have any effect on the value of the objective function if the trajectory does not intersect with any of the obstacle boundaries. Starting with an obstructed feasible trajectory as the initial design vector a neighboring solution, that is locally optimal under the new set of obstacle avoidance constraints, can be found. Figure 13 shows the effect of placing obstacles along the paths of unobstructed low noise trajectories. The resulting solution trajectories are seen to be able to avoid the obstacles imposed without significant penalty to the associated cost function. Lower associated noise levels result if

Figure 11: Optimal approach for initial velocity of 95 knots and final velocity of 45 knots – no acceleration allowed.

Noise-Optimal SNI Longitudinal Approach Trajectories If the additional constraint of obstacles in the longitudinal plane is imposed, the resulting optimal solutions are either local minima corresponding to the unobstructed longitudinal plane or neighboring sub-optimal solutions around the imposed obstacle. The particular choice would depend on the nature of the objective function in the neighborhood of the optimal solution. If the objective function remains relatively constant when perturbed about the optimal solution, the neighborhood sub-optimal

11

the helicopter is allowed to execute a larger number of segments, say 4.

(Fig. 14,15), the resulting trajectories are neighbors to the unobstructed optimal trajectories with the same boundary conditions, resulting in minimal increase in the associated noise level. a)

Figure 13: Effect of Obstacle placement and position on feasible 2-segmented low-noise approaches for constant velocity 70 knots, obtained using gradient descent. b) a)

b)

Figure 15: Effect of Obstacle placement and position on the Optimal Approach for initial velocity 95 knots and final velocity 45 knots.

Figure 14: Effect of Obstacle placement and position on the Optimal Approach for constant velocity 70 knots.

Conclusions A new methodology has been introduced for the automatic generation of SNI trajectories for VTOL aircraft that minimize ground noise. The method has been applied to find trajectories that minimize helicopter BVI noise while avoiding no-fly regions in the longitudinal X-Z plane.

Using the cell decomposition approach to determining minimum noise SNI trajectories, the physical space is first re-decomposed into a modified map of feasible states. But the resulting solution would be globally optimal under the additional constraints of obstacle avoidance. The cell-decomposition approach is designed to generate an obstacle-free search space thus is used to obtain noise-optimal SNI approach trajectories. For both constant speed descents at 70 knots as well as approaches with initial velocity of 95 knots and final velocity of 45 knots

As anticipated, finding trajectories that minimize BVI noise requires the helicopter to operate at large positive or negative values of inflow through the rotor disk. The large inflow forces the shed wake structure away from the rotor blades,

12

reducing the possibility of strong BVI noise radiation. This is achieved by several possible schemes: • If no SNI regions are excluded from the longitudinal plane, the global noise-minimum trajectories consist of a sequence of sawtooth segments at maximum allowable altitude. Accelerated climbs are alternated with decelerating descents reflecting the “bang-bang” nature of the problem. Little time is spent near zer-inflow conditions where BVI noise radiation is a maximum. • A local minimum BVI noise trajectory solution is also found for shallow flight close to the ground. Although the average SEL over the ground plane is minimized at these low altitudes, peak SPL and SEL levels close to the ground will be relatively high. A twosegmented approach, steep decelerating descent to a low velocity, followed by a shallow, very slowly decelerating segment to the landing point is seen to be a good practical approach to minimizing ground noise. • Introducing impenetrable SNI regions in the longitudinal plane force the trajectory to deviate from the unobstructed optimal paths and cause an increase in average SEL levels. It is observed, that under most situations, the resulting sub-optimal paths neighbor the previously obtained optimal solutions and as such continue to radiate relatively low levels of BVI noise. Under some circumstances, it is possible that the presence of obstacles results in a trajectory that is locally optimal for the unobstructed longitudinal plane.

References 1. Newman, D. and Wilkins, R., “Rotorcraft Integration into the Next Generation NAS,” Proceedings of the American Helicopter Society (AHS) 54th Annual Forum, Washington, DC, May 1998. 2. Betts, J.T., “Survey of Numerical Methods for Trajectory Optimization,” Journal of Guidance, Control and Dynamics, Vol. 21, 1998. 3. Seywald, H., Cliff, E., and Well, K., “Range Optimal Trajectories for an Aircraft Flying in the Vertical Plane,” Journal of Guidance, Control and Dynamics, Vol. 17, 1994. 4. Schultz, R.L., “Three-Dimensional Trajectory Optimization for Aircraft,” Journal of Guidance, Control and Dynamics, Vol. 13, 1990. 5. Hagelauer, P., “Contribution a l’Optimisation Dynamique de Trajectoires de Vol pour un Avion de Transport,” Ph.D. Dissertation, CNRS - Universite Paul Sabater de Toulouse, France, June 1997. 6. Slattery, R. and Zhao J., “Trajectory Synthesis for Air Traffic Automation,” Journal of Guidance, Control and Dynamics, Vol. 20, 1997. 7. Brooks, R.A. and Lozano-Perez, T. “A Subdivision Algorithm in Configuration Space for Findpath with Rotation,” in Proceedings of the 8th International Conference on Artificial Intelligence, Karlsruhe, FRG, 799-806, 1983. 8. Russell, S., Norvig, P., Artificial Intelligence: a Modern Approach, Prentice Hall Series, New Jersey, 1995. 9. Schmitz, F. H. ”Reduction of Blade-Vortex Interaction (BVI) Noise through X-Force Control”, NASA TM-110371, April 1995. 10. Gopalan, G., Schmitz, F. H., and Sim, B. W., “Flight Path Management and Control Methodology to Reduce Helicopter BladeVortex (BVI) Noise,” Presented at the American Helicopter Society Vertical Lift Aircraft Design Conference, San Francisco, CA, Jan., 2000. 11. Beddoes, T. S., “A Wake Model for High Resolution Airloads”, International Conference on Rotorcraft Basic Research, North Carolina, February 1985.

The results shown are strongly dependent upon both the noise metric (SELav) and the BVI noise model and should be viewed as trends. SELav does not adequately consider the peak values along the ground, which may be more important than on average level. There are other sources of noise other than BVI noise, which should be included in the noise model and will set the lower levels of noise radiations. These additional sources of noise will raise some of the low levels of SELav that were achieved by the trajectory optimization. However the choice of SELav and the semiempirical BVI noise model used in this paper facilitated the use of optimization methods by keeping the search space tractable in size. Improved noise metrics and noise models must be chosen with similar case.

13

midpoint of the trajectory element and noise levels are propagated to the ground plane using spherical spreading, atmospheric absorption and A-weighting factors. The contribution of this trajectory element to the SELav of the entire trajectory is computed using the expression below:

Appendix A Additional Nomenclature d = Miss-distance at a blade-vortex interaction location = Effective miss-distance for a flight deff condition E = Beddoes wake factor = Inflow Factor fλ K = Indices used for curve-fits reff = Effective impulsiveness measure

χ λi λi,o θTPP ψv ∆ψv

SPLGround (zo , µ, α TPP , θTPP ) = av,dB

Ground Plane

10log10

= Wake skew angle = Induced inflow ratio (positive) = Induced inflow ratio corresponding to zero tip-path plane angle (positive) = Orientation of tip-path plane relative to horizon = Vortex azimuth angle corresponding to a blade vortex intersection location = Wake age corresponding to a blade vortex intersection location

∑



∆t i,n  ∆A n To

  

∆An ∆t n Ao

T

where, SPLdB,,n refers to the sound pressure level received at an observer, denoted by the index n. ∆An represents the area element associated with the observer location. ∆tn refers to the difference between the arrival times of the noise signal from the two end-points of the element. T is the rotor revolution period. Ao refers to the area of the observer plane. The effect of θTPP, the orientation of the tip-path plane angle relative to the horizon, is not an important factor for out-of-plane BVI noise levels, so its effect is ignored. The metric SPLav,dB,Ground is computed for a range of advance ratios, flight path angles and heights z above the ground plane. The error associated with assuming ∆tn to be equal to T was observed to be less than 0.5 dB over the entire range of flight conditions and over the range of z considered in the present study. Therefore, it is assumed that the spatial and temporal dimensions in equation 2 are independent, and the order of the summations reversible, i.e.

F = SEL av =

∑

SPL dB, n

10

n

A.1 Simplified Approximation to the Noise Metric, SELav SELav was defined in Reference 10 as:

Ground  Trajectory Plane  Elements SPL 10  10log10 10 dB,i,n  n i  

∑

Ao

(A.1.1) In the above expression “n” refers to a spatial location on the observer plane and “i” refers to an element along the trajectory. For the current study, a representative ground observer plane 8000 feet by 4000 feet is used for all ground noise calculations. To be able to reverse the order of the spatial and temporal summations in the expression above, a study was conducted, using “Q-SAM” to estimate the effect of ignoring the spatial variation of the ∆t in the computation of average radiated noise levels on the ground associated with a single trajectory element. A trajectory element representing an interval equal to one main rotor revolution of the helicopter, for a given steady flight condition, is oriented in the longitudinal plane at the appropriate flight path angle. The midpoint of this element is placed a height z above the center of the ground plane. For a given advance ratio and flight path angle, the appropriate radiation sphere is oriented at the

Ground  Trajectory Plane  Elements SPL 10  10 dB,i, n  i n  

∑

∑

Trajectory  Ground Elements  Plane SPL 10  10 dB,i, n  n i  

∑

∑

 

∆t i,n ∆A n   

≈

  An    

∆

∆t i

The average SEL over the ground plane can now be expressed as the time-summation of the average SPL over the ground plane associated with each element along the flight trajectory.

F ≈ 10log10

Trajectory Elements

∑ i

(A.1.2)

14

10

SPLGround (z i , µ i , α TPP, i ) 10 av, dB

∆t i To

λ = λ i − µsinα TPP = λ i,0 +

A.2 Derivation of the Blade-Vortex Interaction (BVI) Noise model The current study focuses on the main rotor BVI noise radiated by a two bladed helicopter under nominal longitudinal descent conditions. For specific examples presented in this paper, an AH1 helicopter is used. Out-of-plane main-rotor blade-vortex interaction (BVI) noise characteristics associated with a given rotor design under steady flight conditions, characterized by a constant flight velocity and flight path angle, have been shown to be a function of the advance ratio, tip-path plane angle as well as thrust coefficient CT and the hover tip mach number MH; CT and MH are assumed to remain approximately constant during nominal approach trajectories, and the radiated BVI noise becomes a function of the advance ratio and the rotor tip-path plane angle.

(

∂λ i ∂α TPP 

α TPP − µα TPP + Ο α 2TPP

≈ λ i,0 − µα TPP 1 −  

)

α TPP = 0

1 ∂λ i µ ∂α TPP

α TPP = 0

   

≡ λ i,0 − µα TPP f λ

(A.2) where, µ 4 + C 2T − µ 2

λ i,0 =

2

and  f λ = 1 −  

1 ∂λ i µ ∂α TPP

 =  αTPP =0 

1 2 4 1 + 1 1 + C T µ 2

 

This linearization in tip-path plane angle works reasonable well for tip-path plane angles less than 10°. The tip-path plane that corresponds to zero average inflow through the rotor disk is given by:

Typically, at a fixed advance ratio, as the flight path angle is varied from zero degrees at level flight to steeper descent approach angles, the peak BVI noise level increases to a maximum value and then begins to reduce again. This variation of the peak BVI noise level with tip-path plane angle corresponds to the wake effectively operating below the rotor disk at small tip-path plane angles and shallow flight path angles, cutting through or near the rotor disk at intermediate tip-path plane angles and finally being pushed up above the rotor disk by the inflow field for steep descent flight conditions which correspond to higher tippath plane angles. The condition corresponding to zero average inflow through the rotor disk is usually associated with a high likelihood of strong BVI noise radiation.

α TPP,0 =

λ i,0 

µ1 −  

1 ∂λ i µ ∂α TPP

   αTPP = 0 

=

λ i,0

f λµ

(A.2..3)

In reality, however, several distinct blade vortex interactions occur at each advance ratio and rotor tip-path-plane angle. The BVI noise radiation associated with any individual blade vortex interaction for a given rotor system at a fixed operational thrust coefficient and hover tip Mach number is governed primarily by the impulsiveness associated with the interaction, the vortex strength, and the focusing and defocusing effects of phase addition or cancellation as well as Doppler and dipole effects. At a fixed advance ratio, the basic geometry of each interaction in the plane of the rotor disk is fixed, and the noise radiation associated with any interaction is primarily a function of the average miss-distance during the interaction. Therefore, the peak sound pressure level associated with any individual blade vortex interaction can be estimated as:

From momentum theory, the inflow ratio, defined positive down through the rotor disk, can be related to the induced velocity and the component of flight velocity normal to the rotor disk:

λ = λ i − µsinα TPP ≈ λ i − µα TPP (A.2.1) Linearizing the solution to the momentum theory quartic in forward flight, for small changes in the tip-path plane angle about the zero tip-path plane condition the following form for the inflow ratio can be derived9:

(

SPL ≈ SPL o − 20 log10 d 2

15

eff

)

2 reff +1

(A.2.4)

In the above expression, deff represents the average or effective miss-distance corresponding to the interaction. The term reff represents an impulsiveness factor that determines the gradient of the peak SPL as a function of the average missdistance associated with the interaction. The peak SPL associated with an interaction peaks at the tip-path-plane angle corresponding to zero average or effective miss-distance for that interaction, at the specified advance ratio. Using the modified Beddoes wake model11, the miss distance along any blade vortex interaction location can be expressed as:

- µα TPP +     cosψ v + d = ∆ψ v    λ i 1 + E µ∆ψ v − sinψ  v    2 

where, E = Taking   cos 

an

ψv +

χ

2

χ = arctan

χ ′ = arctan

effective

value

      (A.2.5) 3      

for

the

term



− sinψ v  along any blade 2  vortex interaction, the expression for the average miss distance along an interaction can be expressed as: 3

α TPP,0 (µ ) =

  χ   d ≡ ∆ψ v - µα TPP + λ i 1 + f d   (A.2.6)   2  

λ λ i,0 ≈ − α TPP f λ µ µ

(A.2.7)

λ i,0   π λi,0   f d  1 +  − µ   4 2µ     f λ 1 −  

λi,0

2µ



π



4

f d  − 

−

λi,0 

2µ

f d  



(1 − fλ )  

(A.2.8)

where, 

µ ∆ψ v

 

2

f d =  cosψ v +

and

The small angle assumption for χ′ usually works for an advance ratio greater than 0.1, and as long as the tip-path plane is a relatively small angle, less than 10°. At an advance ratio of 0.1, the term λ i,0 corresponds to an angle of about 10° for the µ AH-1 helicopter. The contribution of the tip-path plane to χ′ increases as the flight path angle becomes more positive. If only descent flight conditions are considered, level flight would correspond to a maximum value for χ′ at an advance ratio of 0.1. The tip path plane angle at this flight condition is less than 1° and its contribution is further reduced by the fact that fλ is slightly less than 1. If it is assumed that both χ′and the tip-path plane angle remain relatively small under all flight conditions considered here, the tip-path plane angle corresponding to zero average miss distance along any individual interaction can be expressed to first order as:

.

µ∆ψ v

µ λ

− sinψ v

3

   

The miss distance associated with any blade vortex interaction, at a given advance ratio can now be expressed, to first order, as:

In the above expression the bars over the terms associated with the wake age and the vortex azimuth represent averaged values for these quantities for any particular interaction. The wake skew angle, χ, is usually defined as the arctangent of the ratio of the advance ratio to the inflow ratio. When the average inflow through the rotor disk becomes close to zero, the absolute value of wake skew angle is about 90°. The parameter χ′ is defined as the arctangent of the ratio of the advance ratio to the inflow ratio, and determines how close the vortex elements get to the blade at the time of the interaction.

d(α TPP , µ) ≈   λ i,0  fd  − f λ 1 − 2µ    ∆ψ v µ  λ  π − i,0 f 1 − f d λ  4 2µ  

(

      

)

(α

TPP,0

- α TPP )

or,

(

d(α TPP , µ) ≈ I1µ α TPP,0 - α TPP

16

)

(A.2.9)

SPL av,dB, total (α TPP , µ) ≈ SPLall0, dBBVI

where 



λ i,0





2µ

I1 = ∆ψ v f λ 1 −



(

 π λ i,0  f d (1 − f λ ) −  2µ  4  

f d  −  

SPL peak, dB (α TPP , µ) ≈ SPL 0,dB

(

= SPL 0,dB − 20log 10 1 + I 2 µ (α TPP,0 - α TPP ) 2

2

d 2eff 2 reff

)

(A.2.11)

2

BVI SPL av,dB,total = SPLall0,dB

   

 − 20  

(A.2.10)

(α (α

) ( )log (1 + I

( µ (α

TPP

BVI BVI < α all log10 1 + I 2,1µ 2 α allTPP,0, - α TPP TPP,0,eff eff

TPP

BVI > α all TPP,0,eff

10

2

2,2

all BVI TPP,0,eff

- α TPP

(A.2.12)

where, I2 =

))

To better approximate the non-symmetric variation of SPLav,dB,total as a function of the operational tip-path-plane angle relative to the effective αTPP,o,eff corresponding to all BVI’s the noise function is modified to include two distinct sets of indices as below:

The peak BVI noise level associated with any interaction at a given advance ratio is therefore a primarily a function of the tip-path plane angle, for small tip-path plane angles, and approximately symmetric about the peak value,  − 20log 10 1 + 

(

BVI − 20log 10 1 + I 2 µ 2 α all - α TPP TPP,0, eff

2 1 2 eff

If the BVI noise levels associated with any steady flight condition are propagated to a representative ground plane a fixed distance z above the ground, such that the ground plane captures a significant portion of the radiated BVI noise, the resulting trends for the area-averaged SPL over the ground plane can also be approximated by the equation above.

I r

I2 is a function of the advance ratio for a specific BVI. This paper uses analytical estimates of the trends of BVI noise levels computed on the surface of a radiation sphere. The average radiated sound power over a one rotor revolution period is computed over a sphere centered at the mid-interval hub location. For a fixed advance ratio, in the range 0.1 to 0.21, equation 10 was observed to correlate well with the trends for peak SPL associated with each individual BVI for the AH-1 two-bladed rotor system. It was further observed that the area-average SPL levels over the entire radiation sphere corresponding to any individual BVI followed similar trends; however the SPL and reff associated with the area-averaged levels are smaller compared to those associated with peak levels. The trends for the total average BVI sound power, associated with all the interactions occurring at a given flight condition can be approximated by summing up the trends for the individual BVI on an energy basis. This process does not account for phase cancellation or addition that may be significant under some flight conditions. Because the radiated noise level associated with each interaction peaks at a slightly different tip-path plane angle, the resultant trend in not necessarily symmetric about the peak value. The equation used for the average radiated sound power trends associated with all BVI is as follows:

Figure A.1: Average radiated BVI sound power over a radiation sphere, 550 feet in radius, as a function of the main-rotor tip-path plane angle.

Appendix B Curve-Fit Procedure The behavior of SPLo and αTPP,0,eff are plotted in Figure B.1. A cubic spline interpolation is used to determine the peak average acoustic power for different advance ratios in the range 0.1 and 0.21,

17

) )+ ) )  2

2

and the corresponding tip-path-plane angles are also established. Po is observed to be a monotonically increasing function in this advance ratio range. A power variation with the advancing tip Mach number, MAT is assumed initially:

SPLallo, dBBVI = K 1 (1 + µ )

K2

The proposed semi-empirical form is curve-fit to the available data. The values of Ki , i = 1, 2, 3 and 4 are then computed using a least squares curve fit optimization procedure, and the correlation is shown in Fig. B.1. The trends corresponding to the curve fits are compared with the analytical data for advance ratios 0.12 and 0.21 in figures B.2 and 3, respectively.

(B.1)

Figure B.2: Curve-fit for the average radiated BVI sound power the ground plane, as a function of the main-rotor tip-path plane angle, for an advance ratio of 0.120. Figure A.2: Average radiated BVI sound power over a ground plane, 4000 feet by 8000 feet, and 500 feet below the helicopter, as a function of the main-rotor tip-path plane angle.

Figure B.4: Curve-fit for the average radiated BVI sound power the ground plane, as a function of the main-rotor tip-path plane angle, for an advance ratio of 0.210.

Figure B.1: Variation of Pav,dB,0 and αTPP,0,eff as a function of the advance ratio.

αTPP,0,eff

can also be approximated semi-

empirically:

α all BVI

TPP,0, eff

≈k

CT

µ

2

≈

k3

µk4

(B.2)

18