1 Longitudinal-Plane Simultaneous Non-Interfering ... - CiteSeerX

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

Min Xue

Ella M. Atkins

Fredric H. Schmitz

Alfred E. Gessow Rotorcraft Center University of Maryland at College Park College Park, MD 20742 T = Main rotor revolution period Abstract ∆t = Elemental time-step Runway-independent aircraft (RIA) operating V = Flight velocity under simultaneous non-interfering (SNI) traffic & V = Acceleration parallel to flight path procedures have been proposed to alleviate W = Helicopter weight airspace congestion at crowded urban airports. x,z = Spatial coordinates This paper introduces a methodology for z = Height of the helicopter above the designing minimum-noise longitudinal SNI ground approach trajectories for rotorcraft. An analytical model for ground noise annoyance associated αTPP = Main rotor tip-path-plane angle with out of plane Blade-Vortex Interaction (BVI) (positive nose up) noise is introduced and its application as the cost χ = Wake skew angle function for SNI trajectory optimization is µ = Advance ratio, V/ΩR described. The noise model relies on a physicsγ = Flight path angle (negative in descent) based semi-empirical expression developed to approximate the average annoyance levels λ = Average rotor inflow ratio (positive associated with BVI noise on a representative for downwash) ground plane. To guarantee strictly SNI λi = Induced inflow ratio (positive) trajectories, fixed-wing traffic corridors are λi,o = Induced inflow ratio corresponding to treated as impenetrable obstacles modeled by zero tip-path plane angle (positive) their cross-sections in the longitudinal approach ψv = Vortex azimuth angle corresponding plane. Two optimization procedures are to a blade vortex intersection location employed: a heuristic strategy that specifies ∆ψv = Wake age corresponding to a blade trajectories using a small approach waypoint set vortex intersection location and a globally-optimal cell-based algorithm. The Subscripts feasibility and practicality of example minimumav = spatial averaging BVI noise solutions are discussed. dB = expressed in decibel n = spatial index over the ground plane Nomenclature i = temporal index or time-step Ao = Ground plane total area associated with the trajectory ∆A = Elemental area on the ground plane o = peak level or reference value a = Acceleration parallel to flight path = Thrust coefficient CT d = Miss-distance at a blade-vortex Introduction interaction location Deff = Effective drag force acting on the The world’s need to travel continues to helicopter put tremendous growth pressure on the F(X) = Objective function commercial airline system. Because few new fλ = Inflow Factor airports are being planned, many airports operate g = Acceleration due to gravity (32.2 at or near capacity with many others approaching ft/sec2) a similar state. Runway-independent aircraft g(X) = Non-linear constraint function (RIA) defined as either VTOL (vertical takeoff I = Indices used for curve-fits and landing) or eSTOL (extremely short takeoff K = Indices used for curve-fits and landing) have been proposed as an = Hover-tip Mach number, ΩR/ao MHT alternative for short to medium range ( α  TPP,0,eff  

The procedure for arriving at the governing coefficients is found in Appendix A. 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. 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

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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. 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 tippath-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. 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 on 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, α TPP ) = SPLGround (zo , µ, α TPP ) av,dB

av,dB

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, 0.18,8 and 0.21.

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.

(6)

+ ∆SPL(z, z o )

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 z above the ground plane center.

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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. 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 conditions become 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.

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 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 = 10log 10

t final

∫

10

SPLGround (z, µ, α TPP ) 10 av, dB

t initial

= 10log 10

dt To

(7)

t final

∫ f(z, µ, α TPP )dt

t initial

where

z& = Vsinγ

,

µ=V

ΩR

and

& D eff V −γ− . W g If it is assumed that the main rotor tip-speed is held approximately constant during nominal approach trajectories, using the relations above, the optimization problem statement can be posed as:

F = 10log10

α TPP = −

≡ 10log10

t initial

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

dx

x initial x final

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

where,

γ≈

dz & ≈ V(x) ∂V(x) and V dx ∂x

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

(8)

where,

and t

z (t ) = zinitial +

&

(9)

&

t initial

V (x ) ) ∫ f(z (x ), V(x ), γ (x ) + g - V(x)

x initial

& (t ) and γ(t) that minimize Find the functions V the function: F = 10log10

x final

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

2

.

∫ Vsinγ dt tinitial

V (t ) = Vinitial +

The term E(x) represents the sum of the kinetic and potential energies associated with the helicopter. The objective function therefore depends only on z(x) and V(x) and their derivatives. While z(x) completely specifies trajectory geometry, V(x) imposes a dynamic character to the trajectory in terms of a velocity profile. It should be noted that the equations of motion of the helicopter along with the associated noise radiation characteristics couple the choice of z(x) and V(x), preempting the possibility of selecting them independently.

t

∫ V& dt tinitial

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 helicopter performance 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

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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 either constant velocity or 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 are further idealized to be piecewise-linear continuous functions along the trajectory:

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. 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 cell decomposition 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.

z i,i −1 (x ) ≈ z i −1 − γ i, i −1 [x − x i −1 ] V2

i, i −1

(x ) ≈ Vi2−1 − 2V& i,i −1[x − x i −1 ]

Trajectory Optimization Using Gradient Descent An n-segmented approach trajectory is assumed to consist of a series of “n-1” waypoints, which, 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:

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 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: n xi

dx

i =1x i -1

i, i -1

F = 10log 10 ∑

∫ f(z i, i -1 (x ), Vi, i -1 (x ), γ i, i -1 , V& i, i -1 ) - V

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

(x )

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

F(X) : Objective Function, Equation 10

V2 - V2 i

i -1

X LB ≤ X ≤ X UB : Lower and Upper Bounds on X

2(x i −1 − x i )

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.

& -V & ,-V & +V & , g(X) = [V i, i -1 max i, i -1 min γ i,i -1 - γ max ,- γ i,i -1 + γ min ] ≤ 0 : Non - linear constraints (Bounds on acceleration 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

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regions, each characterized by one minimum solution obtainable through conventional gradient based approaches to minimization. This bifurcation curve is a locus of the z coordinates that for each value of x correspond to a maximum value of the objective function associated with the resulting trajectory. When climbs to a specified height are allowed, the trajectory begins with a climb to that height and then a steep descent to the final point. As n is increased, the climb segment asymptotes to a smooth curve that starts relatively steep and flattens out before finally entering into the steep descent. 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 may be critical to the success of this method, especially when the design vector becomes large in size. This method is also extendable to three dimensions, at the cost of design vector size. It is to be noted that because of the approximations made in arriving at the analytical 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. The SEL distribution corresponding to the optimal trajectory with the lowest value of SELav, could include small regions with relatively high SEL levels. Therefore, depending on the noise sensitivity distributions over a specific ground plane, different optimal or close to optimal solutions may be more desirable or acceptable.

non-unique locally optimal solutions under nominal approach conditions. By varying the initial value of the design vector Xo, potentially several local minima can be found and the trajectory corresponding to the lowest value of the objective function is chosen. 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+1segmented trajectory are generated. This process is repeated till a trajectory with an acceptable value for the objective function is obtained. The globally minimum value of the objective function for an order-n+1 trajectory cannot be greater than that of an order-n trajectory. 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 constraints. In effect, this would yield the most acoustically efficient trajectory defined by “n” waypoints. This procedure was first applied to constant speed trajectories. The boundary values were chosen as x = 0 ft, z = 50 ft and x = 50,000 ft, z = 1000 ft and the velocity was assumed to be 70 knots (Fig. 4). The z dimension was bound by 50 ft and 2500 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 waypoints actually yields a feasible solution. It turns out that this solution is not only feasible but also close to one of the local minima of this problem. By placing one waypoint along this trajectory, n is increased to 2, and the optimization procedure is performed. The results are presented in Fig. 8. The solution converges to a very similar trajectory with a flight path angle close to -1°, with a characteristic SELav of about 86 dB. By choosing other feasible initial conditions, another minimum solution is obtained, with an SELav of about 78 dB (Fig. 7). This trajectory starts with level flight followed by a 9° descent into the endpoint. A bifurcation curve z=b(x) divides the x,z domain of feasible initial solutions into two

Figure 5: Several low-noise solutions for a 2segmented trajectory obtained using gradient descent.

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Fig. 8. Additionally, because a rotorcraft climbs/descends with flight path angle between ± 9o , 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°.

The maximum noise trajectories corresponding to 4.8 degree approaches are also shown for comparison.

(Figures 6 and 7 to be put in and talked about !!!)

Trajectory Optimization using CellDecomposition The BVI noise model is adopted as the cost function during final approach trajectory generation. The goal is to identify globallyoptimal SNI trajectories given realistic constraints. An approximate cell decomposition method [9] 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: 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.

Figure 8: Examples of Approximate Cell Decomposition and Modified Approximate Cell Decomposition

Once the modified approximate cell decomposition map is created, this space must be explored to identify the optimal trajectory given boundary condition pair (xi,zi,Vi) and (xf,zf,Vf). Typical approaches include dynamic programming and A* search [10], 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

A quad-tree data structure was used for cell decomposition. At every depth level of the quad-tree, 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. 8 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

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thus the trajectory optimizer utilizes uniformcost search with cost g(n) set to radiated ground noise summed over the path from initial state to node n.

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 final position is x = 1,500 feet and z = 50 feet. Solution trajectories corresponding to constant speed approaches are identical to those obtained using the gradientdescent method. This indicates that the globallyoptimal solution for constant speed approaches is actually composed of a small number of segments. For the same initial and final velocity boundary conditions(70 knots), the helicopter must 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. 9), the resulting optimal 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 allowed to climb (Fig. 10), the helicopter flight path is an alternating sequence of steep climbs and descents, with climb accompanied by acceleration and descent 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.

Figure 9: Optimal approach trajectory for initial and final velocities of 70 knots – no climb allowed..

Figure 10: Optimal approach trajectory for initial and final velocities of 70 knot with climbs allowed. A helicopter in 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. 11) but not climb or accelerate. The corresponding optimal trajectory starts with a steep

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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. 12), the basic trajectory profile of Fig. 11 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. 13), 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. 11 and 12. This global optimum solution is compared with the “worstnoise” approach, a shallow descent approach that always maintains the inflow at low value. All other solutions corresponding to this set of boundary conditions are constrained minima and as such are sub-optimal when the artificial climb and acceleration constraints are relaxed.

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

Figure 12: Optimal approach for initial velocity of 95 knots and final velocity of 45 knots – no acceleration allowed.

Figure 13: Optimal Approach for initial velocity of 95 knots and final velocity of 45 knots -- allow deceleration, acceleration, climb and descent.

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

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function a neighboring sub-optimal solution can be found. This method would work in general if the resulting solution is “close” to the initial solution. Using cell decomposition, the physical space would need to be re-decomposed into a modified map of feasible states. But the resulting solution would be globally optimal under the additional constraints of obstacle avoidance. Thecell-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 (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.

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 suboptimal 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 constraints to the helicopter’s ability to climb or accelerate along the flight path, and select the trajectory that is not obstructed by the imposed obstacles. If no such trajectory exists then determine neighboring suboptimal solutions that may still be acceptable. This can be done in two ways. Using gradient descent, starting with the obstructed optimum solution as the initial design vector and imposing the obstruction as a suitably defined potential

b)

b)

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

ADD 1-2 SENTENCES HERE – THIS SECTION ENDS ABRUPTLY OTHERWISE. WHY DID YOU CHOOSE THESE OBSTACLES PLACEMENTS? CAN YOU EXTRAPOLATE TO OTHER CASES – MULTIPLE OBSTACLES, ETC? ADD A SENTENCE ABOUT HOW LONGITUDINAL TRAJECTORY DESIGN ULTIMATELY BREAKS WITH TOO MANY OBSTACLES (REQUIRING A SOLUTION OUTSIDE THE GAMMA CONSTRAINTS, FOR EXAMPLE).

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

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.

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. 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, 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 saw-tooth 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 zero-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 two-segmented approach, steep decelerating descent to a low velocity, followed by a shallow, 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 common situations the resulting

Optimal approach trajectory results are strongly dependent upon both the noise metric (SELave) and the BVI noise model and should be viewed as trends. SELave 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 SELavg that were achieved by the trajectory optimization. However the choice of SELave 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.

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.

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6.

7.

8. 9.

10.

11.

12. 13.

Dissertation, CNRS - Universite Paul Sabater de Toulouse, France, June 1997. Slattery, R. and Zhao J., “Trajectory Synthesis for Air Traffic Automation,” Journal of Guidance, Control and Dynamics, Vol. 20, 1997. 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. Russell, S., Norvig, P., Artificial Intelligence: a Modern Approach, Prentice Hall Series, New Jersey, 1995. 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. Beddoes, T. S., “A Wake Model for High Resolution Airloads”, International Conference on Rotorcraft Basic Research, North Carolina, February 1985. Ffowcs-Williams, J. E. and Hawkings, D. L., “Sound Generation by Turbulence and Surfaces in Arbiturary Motion”, Phil. Trans. Roy. Soc., A264, pp. 321-342, 1969. Lowson, M. V., “The Sound Field for Singularities in Motion”, Proc.Roy.Soc. A, Vol. 286, pp. 559-572, 1965. Schmitz, F. H., Gopalan, G. and Sim, B. W., “Flight Trajectory Management to Reduce Helicopter Blade-Vortex Interaction (BVI) Noise with Head/Tailwind Effects”, presented at the 26th European Rotorcraft Forum, The Hague, The Netherlands, September 26-29, 2000.

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λ = λ i − µsinα TPP

Appendix A = λ i,0 +

Derivation of Blade-Vortex Interaction 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 AH-1 helicopter is used. Out-ofplane 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. 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 tip-path 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. 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 [15]:

∂λ i ∂α TPP

(

α TPP − µα TPP + Ο α 2TPP

)

α TPP = 0

 1 ∂λ i ≈ λ i,0 − µα TPP 1 −  µ ∂α TPP 

  ≡ λ − µα f i,0 TPP λ  α TPP = 0  (A.2)

where,

λ i,0 =

µ 4 + C 2T − µ 2 2

and  1 ∂λ i f λ = 1 −  µ ∂α TPP 

 =  α TPP =0 

1 1 + 1 1 + C 2T µ 4   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: α TPP,0 =

λ i,0  1 ∂λ i µ 1 −  µ ∂α TPP 

   α TPP = 0 

=

λ i,0 fλµ

(A.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:

λ = λ i − µsinα TPP ≈ λ i − µα TPP (A.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 derived:

15

SPL ≈ 20 log 10

P(n, µ) 2

d eff r

(

2 eff

+1

)

2 = SPL o − 20 log 10 d 2eff reff +1

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

(A.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 miss-distance 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 model [12], 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  - µα TPP +     cosψ v + = ∆ψ v    λ i 1 + E µ∆ψ v − sinψ  v    2  (A.5)

       3     

  χ   = ∆ψ v - µα TPP + λ i 1 + f d   2      (A.6) where,

 µ ∆ψ v f d =  cosψ v + − sinψ v  2 

3

   

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.

      3      

      3      

µ and λ λ λ i,0 χ ′ = arctan ≈ − α TPP f λ µ µ

χ = arctan

where, E =

χ 2

.

Taking an effective value for the term µ∆ψ v  3 − sinψ v  along any blade  cosψ v + 2   vortex interaction, the expression for the average miss distance along an interaction can be expressed as:

(A.7)

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 λ i,0 corresponds advance ratio of 0.1, the term µ to an angle of about 10° for the 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

16

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:

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 areaaverage 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:

λ i,0   π λ i,0   f d  1 +  − µ   4 2µ   α TPP,0 (µµ =   λ i,0   π λ i,0   f d (1 − f λ ) f d  −  − f λ 1 − 2µ   4 2µ    

(A.8) The miss distance associated with any blade vortex interaction, at a given advance ratio can now be expressed, to first order, as:

d(α TPP , µ) ≈   λ i,0   f d  −  f λ 1 − 2µ     ∆ψ v µ   (α TPP,0 - α TPP ) λ  π − i,0 f (1 − f ) λ  4 2µ  d    or, d(α TPP , n, µ) ≈ I1µ (α TPP,0 - α TPP ) (9)

where   λ i,0   π λ i,0 fd  −  − I1 = ∆ψ v f λ 1 − 2µ   4 2µ  

  f d (1 − f λ )  

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

(

(

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

2

)

)) 2

(A.12)

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,

 d2 SPL peak,dB (α TPP , µ) ≈ SPL 0,dB − 20log 10 1 + eff 2 reff 

(

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

To better approximate the nonsymmetric 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:

   

SPL av,dB,total = SPLall0,dBBVI

( (

) )

( (

( (

 α < α all BVI log 1 + I µ 2 α all BVI - α TPP 10 2,1 TPP TPP,0, eff TPP,0, eff − 20  all BVI 2 all BVI  α TPP > α TPP,0,eff log10 1 + I 2,2 µ α TPP,0,eff - α TPP

(A.10) where, I12 I2 = 2 reff

(A.13)

17

) )+ ) )  2

2

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.

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.

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.

18

= 1, 2, 7 and 8 are then computed using a least squares curve fit optimization procedure.

Appendix B Curve-Fit Procedure The equation used for the average SPL trends associated with the BVI noise radiated over either a radiation sphere or a representative ground plane a fixed distance z below the helicopter, was expressed in Appendix A as follows: SPL av,dB, total (α TPP , µ) ≈ SPLall0, dBBVI

(

(

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

Index K1 K2 K7 K8

)) 2

Radiation Sphere Trends, R = 550 ft. 81.1 dB 0.55 0.082 2.12

Ground Noise Trends, Z = 500 ft. 61.9 dB 1.04 0.079 2.13

Table B.1: Indices representing the variation of Po and αTPP,o with advance ratio for both radiation sphere trends and Ground noise trends.

(B.1) To better approximate the nonsymmetric variation of SPLav,dB,total as a function of the operational tip-path-plane angle relative to the peak value, the noise function is modified to include two distinct sets of indices as below:

SPL av,dB,total = SPLall0,dBBVI

(

BVI  α TPP < α all TPP,0, eff  log 1 + I µ 2 10 2,1 − 20 all BVI α > α TPP,0,eff  TPP log 1 + I µ 2 2,2  10

(

(

(

)× (α )× (α

all BVI TPP,0, eff

  +     

- α TPP

))

- α TPP

))

all BVI TPP,0, eff

2

2

(B.2) Figure B.1: Variation of Pav,dB,0 and αTPP,0,eff as a function of the advance ratio.

The functional behavior of SPLo and

αTPP,0,eff are first established. These trends are

I2 was expressed as a polynomial/power function of the advance ratio. These indices are plotted as a function of advance ratio in Figure B.2.

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, 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 ) αTPP,0,eff

K2

(B.3)

can also be approximated

semi-empirically: BVI α all ≈k TPP,0,eff

k CT ≈ k87 µ2 µ

(B.4)

Figure B.2: Variation of the index I2 as a function of the advance ratio.

The proposed semi-empirical form is curve-fit to the available data. The values of Ki , i

19

The trends corresponding to the curve fits are compared with the analytical data for three different advance ratios in figures B.1-3.

Figure B.3: 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 B.5: 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.

20