Variable Speed Transmission using Planetary Gear System for High Speed Rotorcraft Application Sylvester V. Ashok
[email protected] Brian Wade
[email protected] Daniel P. Schrage
[email protected] Georgia Institute of Technology, Atlanta, GA 30308 ABSTRACT High-speed helicopter flight poses many challenges to designers. Many of these challenges result from the additive nature of the helicopter speed and the rotor rotation on the helicopter’s main rotor advancing side. These high speeds result in increased wave drag, impulse noise and vibrations. These disadvantages can be lessened by slowing the main rotor during high speed flight. This paper presents a transmission design that slows the main rotor by up to 50% during highspeed forward flight while maintaining high rotational speed during low-speed flight and hover and maintaining engine efficiency. The transmission design uses a standard planetary gear system with the sun gear as the input and the planetary carrier as the output. It accomplishes its speed variation by allowing the ring gear to rotate at variable speeds controlled by a controller and an electric motor. This design was demonstrated on a theoretical coaxial helicopter. This paper also covers the sizing and synthesis methodology used to perform preliminary design of this transmission.
INTRODUCTION The main rotor is the most complex part of a helicopter. In forward flight it poses further challenges due to differential tangential velocities observed by the advancing and the retreating sides. This is further exacerbated at high velocities where the tip speed at the advancing side is transonic and retreating side is near stall. Cyclic feathering helps to alleviate the differential lift, but the maximum speed of the helicopter begins to be limited by the speed of the advancing rotor blade tip. As the advancing blade enters the transonic regime, it develops wave drag that dramatically increases the power required from the engine. As this wave drag begins to form, vibrations also increase due to shock induced and dynamic stall and the resulting shocks greatly increase the in-plane impulse noise of the blade. A way to delay these effects is to slow the main rotor rotational speed. Current rotorcraft designs slow the main rotor through the engine, forcing the engine to operate off-design and results in degraded efficiency. This method, however, is contrary to what is needed in hover and slow flight. In a hover, the rotor does not have the advantage of forward speed. As the rotor speed is slowed, the blades must be pitched to higher and higher angles of attack until they stall. Typically, designers seek to compromise between the hover and Presented at the American Helicopter Society 66th Annual Forum, May 11-13, 2010, Phoenix, AZ.. Copyright © 2010 by the American Helicopter Society International, Inc. All rights reserved.
forward flight performance choosing a tip speed that is fast enough to allow the helicopter to maneuver at a hover with a safety margin before stall yet slow enough to allow the helicopter to reach its required forward speed.
Figure 1. Aerodynamic, noise, and autorotative constraints imposed on selection of rotor tip speed [2] Figure 1 (above) shows the optimum tip speed consideration for a given forward flight speed limit, in order to expand this operating limit, one has to maintain the optimum tip speed while achieving higher advance ratios.
Understanding the above limitations, it seems reasonable that a helicopter would be best served by allowing the rotor tip speed to be changed in flight from high speed at a hover to low speed in forward flight. This can be done to an extent by slowing the engine, as done in present designs. However, this speed change is restricted by the operating limits of the engine. All modern turbine engines are designed around a specific operating speed. Outside this normal design speed, the efficiency falls drastically to the point where many engines cannot deliver enough power to fly the helicopter at the proposed 50% speed reduction of this paper. Another alternative to changing the speed of the engine is to slow the output shaft from the main transmission. The two categories of transmissions that do this are continuously variable transmissions (CVT) and multispeed transmissions. CVTs can operate at any speed between a minimum and a maximum speed while multispeed transmissions operate at a finite number of speeds. The conventional two-speed designs are less complex and do not have as great of a weight penalty as the CVTs, but they posses inherent power interruptions during speed change and negative power flow. Sudden interruptions in power can develop severe structural loads on the transmission and interruptions in power flow will cause loss in performance. Two-speed designs can be adaptable to be quasi-variable through variable transition assist, which may be either external powered or internally driven. Speed range changes for CVTs need to be computer controlled, sensing both drive speed and engine speed. The design objectives therefore for the transmission is to achieve the speed while maintaining continuous positive power flow without a significant weight penalty. Other design criteria
to be considered are reliability, system and sub-system feasibility, life-cycle cost, total volume and part count. When choosing the speed range needed for the slowed rotor, the rotorcraft performance and the rotors dynamics and handling qualities need to be considered. Slowing the rotor too much will lower the autorotative capabilities and affect the rotors damping. The slowed speed should be chosen such that the rotor motion’s natural frequencies are away from the harmonics of rotation (1P, 2P, etc.). This puts operating constraints on the motor to control the speed change and on the optimum speed change rate, for both slowing and returning to normal operation speed. The motor also needs to constantly power the controller in order to maintain a speed if the desired slowed rotor speed is not slowed by exactly 50%.
CONCEPT OVERVIEW This transmission design consists of a planetary gear with sun gear as input and planetary carrier as output as shown in Figure 2 (below). The planet gears behave as intermediate gears similar to that in a conventional planetary system. The ring gear is held stationary under normal operation, and the planets rotate about it. The normal operation for this design is categorized as hover, climb and low speed forward flight (below bucket speed). The planetary carrier angular velocity is given by the relationship in equation (1)
S: Sun - Input P: Planet – Intermediate gears Planet Carrier: Output Ring: Intermediate gear Controller: Secondary input
Figure 2. System concept and rotation direction
Ns Nr pc s r Ns Nr Ns Nr
Where, (1) Subscript
: Angular velocity N : Number of teeth s: sun r: ring gear pc: planetary carrier
The ring gear speed, ωr is negative, if rotating in the opposite direction to the planetary carrier as shown in Figure 2 (above). ωr term in Equation (1) will now be acting to reduce ωpc. The design concept follows the principle that the planetary carrier angular velocity, for a fixed gear ratio has a dependency on the sun gear (primary input) and the ring gear (intermediate gear for secondary input). The ring gear, shown in Figure 3 (below), is therefore designed with internal gearing allowing it to mesh with the planet gears and outer gearing allowing it to mesh with the controller gear.
Figure 4. Clutch system
Figure 3. Planetary gear system – detailed view A planetary gear offers the advantage of high speed reductions within a small space. It is adaptable to be a CVT with a motorized controller assist for speed transition. This is the fundamental part of this type of a CVT. This mechanism should be adaptable to a drive system of most helicopters. Rotors are designed for operation at particular speeds and this system provides the capability of having a smooth transition between the required multi speeds. This design uses a set of clutch bands meant to hold the ring gear stationary when required and for this purpose the ring gear is provided with extended portions on either side on the outer surface. The clutch works primarily as a holding mechanism. This mechanism is similar to that used in traditional automobile automatic transmissions. A small amount of friction for braking is expected during the slowing down and engagement sequence (this braking frictional force is yet to be quantified). These clutches are designed to be power disengaged, meaning the electric actuators are spring loaded and power has to be supplied to release them. For safe helicopter operation, the clutch system remains engaged without power supply to the actuators. Actuator pistons are inserted within a spring. When the actuators receive electric signal, they compress retracting the bands, this causes the clutch bands to release the ring gear, which is then free to rotate. System is designed with multiple clutch bands and actuators for high reliability and better frictional force distribution as shown in Figure 4 (below).
A control mechanism is required for this system to enable the speed change process. It is the most important requirement in this transmission design. The functions of the controller are to: 1. Control the speed of the ring gear after clutch disengagement. Rotor dynamics and power flow require that the ring gear start rotating slowly and increase speed at a rate optimized for rotor performance. 2. On conclusion of high-speed (flight) operation, the rotor is required to return to its normal speed. The controller is required to slow down the ring and bring it to a safe clutch engagement speed. The controller can be internally driven or externally powered. An internal drive mechanism is one that uses mechanical power flow from the drive system to control the ring gear, Figure 5(a) (below). An external drive mechanism is one in which a motor, powered externally is used to perform the speed/resistance control, Figure 5(b) (below).
(a)
bled off the drive system directly by the use of the variator mechanism. Once the operating speed is reached, the ring gear acts as an output of the planetary system, the variator can be designed to take advantage of this by regenerating that power back to the drive train or to an auxiliary thrust system. The challenges with this system however, is that variator involves complex shafting and gearing, which increase the weight and the volume of the system. The externally driven system is simple, with sufficient gear ratios for high torque, a sizeable motor can be adopted to power the system. This will however have a weight and volume penalty of the motor itself and to provide adequate gearing for torque generation. (b) Figure 5. (a) Internally driven control mechanism; (b) Externally driven control mechanism [1] Each method of control mechanism has its advantages and disadvantages. The internal driven mechanism has the advantage of power regeneration, i.e. the power required by the controller to let the ring gear start rotating at a slow rate, when the rotorcraft is entering high speed mode, is
IMPLEMENTATION OF DESIGN This transmission design was created as part of the Georgia Institute of Technology’s graduate level entry into the 2009 AHS Student Design Competition sponsored by Agusta-Westland to design a unique rotor drive system. The entry, named the Peregrine, won the first place and is shown in Figure 6 (below).
Figure 6. Peregrine – high speed coaxial rotorcraft concept Transmission Layout The transmission system for the Peregrine is split into sections A to E and the gears divided as G1- G12 as shown in the transmission layout in Figure 7 (below). The rotor drive system for this entry featured two standard
turboshaft engines, which feed into a mixing spur assembly (system A in Figure 7). The power was then sent forward into the unique dual speed planetary system (system B in Figure 7) and rearward through bevel gears systems (systems D and E in Figure 7) to a pusher propeller. The output from the planetary system was then
fed into a differential system for reduction and speed matching of the counter rotating rotors (system C in Figure 7). This is done with the help of a hollow shaft which powers the lower coaxial rotor from the upper bevel gear and an inner shaft that powers the upper rotor from
the lower bevel gear. This system has no mechanical dissymmetry in power supply to both rotors i.e. both rotors operate at the same rpm at all operating conditions and the same torque when no differential collective is applied.
Figure 7. Peregrine transmission layout
performance at various operating conditions. This system produced a helicopter capable of 222 knot cruise and 250 knot dash speeds as seen in the Figures 8(a) and (b) (below). Standard sea level conditions did not strike as a big challenge for hover power requirements, as seen in Figure 8(a).
Peregrine Performance Performance analysis was done on the peregrine helicopter to study the effects of transition velocity on forward flight performance of the helicopter. Transition velocity was chosen as 100 knots, with a good cushion on
Max dash speed 250 knots at IRP Max cruise speed 222 knots at MCP
(a)
(b) Figure 8. Required horsepower and available horsepower vs. velocity curve for (a) sea level and (b) high - hot day – 5000ft, 95°F The solid red line shows the actual helicopter power curve including the rotor speed reduction. The blue dash line shows the power for the helicopter at a low speed setting only and the yellow dashed line shows the performance at high speed only. The value of the rotor tip speed reduction is shown from maximum speed at both max continuous power (MCP) and at intermediate rated power (IRP). At sea level standard conditions, the dash speed moved from 210 knots to 250 knots. This 40 knot increase was due only to the rotor speed change that took place in the transmission with the multispeed planetary system. The operation of the planetary system is expressed in Table 1 (below). Table 1. Peregrine transmission settings Mode High Speed Low Speed
Tip Sped Input Output 650 ft/s Sun Gear Planet Carrier 420 ft/s Sun Gear Planet Carrier
Ring Gear Speed 0 – Locked 391 RPM
Rotor Dynamics A fan plot for this design was generated using DYMORE to predict the dynamic characteristics of the rotor. DYMORE was selected to estimate structural deformation and dynamic response such as displacements, strain, forces, and velocity. DYMORE is a finite element based tool for the analysis of nonlinear flexible multibody
systems. To predict the motion of the rotor, DYMORE needs the data of aerodynamic forces, structural properties, rotor motion, and kinematical constraints. This analysis provides the distance between upper and lower rotor systems. In addition, the natural frequency of blade system can be analyzed by the eigenvalues. Since each rotor system has four blades, 1P, 3P, 4P and 5P should be avoided and design emphasis was placed on reducing potential harmonics in this region. In the case of 1P, it does not cause any problem at all, because the first lead lag mode is well above 1P (stiff in plane) for removal of ground and air resonance instabilities. However, Peregrine is a rotorcraft with variable rotor speed main rotor. During transition from a rotor speed of 650 ft/s to 420 ft/s, the initial design had the second flapping mode and the first lead-lag mode pass through the regime of 4P. Furthermore, the frequencies of second flapping and first lead-lag mode were very similar in the low speed mode. This can cause coupling vibration. Both of these situations can result in vibrations and instabilities in the rotor system. Therefore, the blade was stiffened by adjusting the position and shape of spar. Figure 9 (below) shows the fan plot for the improved blade. In this design, the frequencies of second flapping and first lead-lag mode are always higher than 4P. In addition, the coupling vibration was also solved by separating the frequencies during all normal speed regimes.
Figure 9. Fan plot for Peregrine helicopter
Sizing and Synthesis A genetic algorithm was used to size and optimize the gear system to yield the lightest weight system that was able to withstand all gear stresses of the system. All gears were initially selected to be helical type gears with 25 0 full tooth loading due to their gradual loading, quieter running
noise, and capability for high speed, high load operations. Even with the helical and loading angles selected the transmission design outlined above still had 25 independent variables. These variables are listed in Table 2 (below). The gear ratio and diametrical pitch (Pd) apply to the gear set (A-E) and the material selection is done independently for each of the 12 gears.
Table 2. Transmission design variables Pd = diametric pitch, number of teeth per inch, for each gear Pd=[1, 1.25,1.5,1.75,2,2.5,3,4,5,6,8,10,12,14,16,18] Gear Ratios Engine Gear Box: 1 – 3 System A: 1-10 System B: 1-10 System C: 1-4 System D: 0.2 – 5 System E: 0.2-5 Number of planets System B: [2,3,4] Material for each gear Material = [AISI 9310, VASCO X2M, Pyro-wear 53]
These parameters were selected based on knowledge of other gear designs. The diametrical pitch (Pd) set listed is the most commonly manufactured Pd’s for course gears that are used in aircraft transmissions. The gear ratio ranges were selected based on the type of gears used in the system. Spur gears typically can withstand ratios up to 10:1 and bevel gear systems can withstand ratios up to 1:5. System C was limited to 1:4 due to the high loads it would see since it was the last stage of reduction to the rotor. The three material selections are common gear
Number of Variables 5 2 1 1 1 1 1 1 12
materials used in the main transmission. All three materials are steel alloys that are case-hardened and carburized. Their material properties are shown in Table 3 (below). For sizing, the following design criteria were selected: Engine operated at 3100 HP (3442 Contingency Power), 23,000RPM (plus a 20% factor of safety). Rotor tip speed: 680 ft/s in High Mode, 420 ft/s in Low Mode.
sizing since, given the number of variables there was a very high likelihood of many local optimums and 18 of the 23 variables were discrete by their very nature (Pd of each system, material of each gear, and number of planets in system b.)
Transmission component life: 3,500 flight hours Table 3. Gear material properties [5]
Transmission Sizing
Even with these constraints and discritizing the continuous variables to the hundredths there were over 4.598 (10)34 possible combinations. A computer that could analyze 1,000 of these combinations a second would take 1.458 (10)24 years to analyze all the possible combinations (full factorial analysis). A faster optimization method was needed. In many typical designs an initial starting point is determined based on engineering judgment and the parameters are optimized from that point. The difficulty with this method is when dealing with complex and new technologies as with the speed controller proposed by this paper. Often adequate design knowledge does not exist for such emergent technologies. When optimizing from any initial point the optimization process typically settles into the nearest local optimum which may, or in many cases, may not be the global optimum. Therefore, in order to optimize all these variables to create the lightest weight system that would meet the stress requirements of the new system, a genetic algorithm was used. A genetic algorithm (discussed in detail in appendix section) is a stochastic search method that is specifically designed for discrete variables and because it is stochastic it has an increased probability of finding a global optimum in the presence of many local optimums. This was very fitting for the transmission
A sizing program was written for the new transmission in Matlab. This sizing program sized each gear mesh system (labeled A – E in Figure 7). The equations and gear factors used followed the recommend practices of the American Gear Manufactures Association (AGMA) as outlined in [6]. The program’s general design was object oriented in nature. Each gear system (labeled A – E in Figure 7) was sized in its own program and the parameters passed to the main program. In order to ensure the rotor system and propeller each spun at the required rpm the gear ratio of systems B and E were calculated given the input gear ratio of the rest of the gears. This left 23 independent variables. Within each gear mesh sizing program, the general flow was to step through the number of teeth of the pinion (up to a max of 200 teeth), calculate the number of teeth of the gear, and then for each tooth set, increased the face width until the bending and contact stress were less than the allowable stress. The program then estimated the weight for that gear set. This was done for every tooth combination and the lowest weight gear’s parameters (teeth of the pinion and gear, face width, diameters of each gear, and stresses of each gear) were passed back to the main program which then added up the weight of the entire system. Thus, the sizing program was able to optimize the numbered teeth, diameter, and face width of each gear for the lowest overall weight given the 23 inputs. The genetic algorithm was run as specified in the above section using Model Center. Model Center is a graphical user environment to perform integration and automation of codes, optimization and for experimental design. Model Center was used to automate the genetic algorithm and the sizing program. A multiple elitist strategy with tournament selection helped to ensure the appropriate amount of exploration while speeding up convergence to the optimum. A population of 60 was chosen with 15 consecutive generations with no change for the top member as the convergence criteria. The other model parameters for the actual transmission genetic algorithm are summarized in Table 4 (below).
Table 4. Transmission sizing genetic algorithm parameters General Parameters Population Convergence Size Criteria 60 15 generations Constraints Gear 5 – Ring Gear 7+8 – Rotor Bevel Gears 6 – Input to rotor bevel gears
Variable Type Continuous Discrete
Probability of Cross Over 100 100
Constraint Max Diameter = 22 in Max Diameter = 30 in Max Diameter = 10 in
Probability of Mutation 0.1 0.05 Reason Engine air intake obstruction Fuselage width – Drag reduction Fuselage height – Drag reduction
Another benefit to using the multiple elitist strategy is that the cross over probability can be 100%. This means that the bottom 2/3 of the designs is guaranteed to change but the best designs are preserved in the top 1/3 elite. This
also helps to speed up the convergence of the program and helps the program to be exploitive versus exploratory in order to find the best optimum while ignoring other local optimums.
Figure 10. Genetic algorithm optimization convergence history
The results of the genetic algorithm produced an optimum gear set for the above parameters. The program took 113 generations to converge and built approximately 6.24 million transmissions in the process. The convergence history is shown in Figure 10 (above) for the final sizing run. The parameters for this gear set are shown below in tables 5 and 6 (below). As can be seen
from these parameters the constraints in gears 6, 7, and 8 are active (shown in yellow) and gears systems A and C were sized primarily on contact stress while the rest were sized based on bending stress (shown in yellow). Also, note that there is an inherent 20% factor of safety in all actual stress estimates which was built into the sizing program.
Table 5. Optimized Gear Parameters
System Mixing Spur Gears North Planetary System Bevel Gear System South Bevel 1 South Bevel 2 Eng Acc Gear Box
System ID
System Parameters Mach Pd Adv (1/in)
Weight (lbs)
A
3.9048
8
32.33
B
4.1053
8
37.29
C
3
3
154.93
D
1.0968
10
25.15
E
1.3222
10
33.536
Eng
1.628
No. of planets
4
Gear 1
D (in) 2.625
2 3 4 5 6 7 8 9 10 11 12 Eng Acc
10.25 4.75 5 14.75 10 30 30 9.3 10.2 9 11.9
Gear Parameters Face Teeth Width (in) 21 82 38 40 118 30 90 90 93 102 90 119
6.8
2.8
3.9 2.7 3.1
Material Pyro-Wear 53
Weigh t(lb) 3.04
Pyro-Wear 53 VASCO X2M AISI 9310 VASCO X2M VASCO X2M VASCO X2M VASCO X2M VASCO X2M VASCO X2M Pyro-Wear 53 VASCO X2M
29.52 3.33 5.09 13.77 16.81 60.65 60.65 11.45 13.7 12.34 21.2
VASCO X2M
Table 6. Gear Bending and Contact Stress System Parameters System System ID Mixing Spur Gears
A
North Planetary System
B
Bevel Gear System
C
South Bevel 1
D
South Bevel 2
E
Gear Parameters Gear RPM 1 14128 2 3634.6 3 3634.6 4 3452.9 5 0 6 885.36 7 295.57 8 295.57 9 3634.6 10 3304.6 11 3304.6 12 2500
Computer Aided Design model A parametric computer aided design (CAD) model for this design was created in CATIA which enabled rapid
Contact Stress (psi) Actual Allowable 198,940 200,230 100,190 206,580 197,490 200,100 192,490 203,550 63,599 206,700 211,110 211,180 211,110 213,150 211,110 213,150 198,240 207,720 198,240 208,170 207,700 209,510 207,700 209,510
Bending Stress (psi) Actual Allowable 42,592 43,621 38,183 44,688 43,418 43,599 42,964 44,181 40,077 44,709 39,083 41,344 39,083 41,639 39,083 41,639 40,572 41,815 40,572 40,885 40,489 40,885 40,489 41,088
resizing of the transmission to evaluate constraints on individual gear systems dimensions, based on rotorcraft’s structure for each iteration. The final design is shown in Figure 11 (below).
Figure 11. Final transmission design (CAD model) The CAD model enabled some initial finite element analysis (using SIMULIA) to verify theoretical stress
calculations. Figure 12 (below) shows an image of stress analysis performed on the planetary gear system.
Figure 12. Finite Element Analysis on planetary gear system (planet and sun gears shown)
CONTROLLER ARCHITECTURE A Power Electronics Module (PEM) was designed to control the variable speed planetary transmission. This system is shown in Figure 13 (below). The Pilot has control over the selection between high and low speed transitions in this initial concept. At hover and low speed operation, the PEM is set for high rotor rpm, meaning the ring gear clutch bands are engaged. For high-speed flight, the PEM is set for low rotor rpm. The PEM is made up of a Supervisory Controller (SC), Motor Controller (MC) and Clutch Controller (CC). SC monitors controller gear speed and rotor rpm (from the speed governor). The governor works as a primary input to both the engine control system (FADEC – Full Authority Digital Engine Control) and the PEM. The speed switch should have an input to the FADEC system as well to ensure that the engine does not try to reset the speed of the rotor. SC monitors motor torque, to enable accurate inputs to the MC. The MC powers the motor and supplies precise voltages to ultimately keep the reference rpm stable. The CC controls the engagement and disengagement of the clutches. This system should have a provision for manual override from the pilot to be able to engage the clutches for flight safety purposes. This is indicated by the dashed line in Figure 13. The PEM should also receive an input from the vehicle electrical system.
High rpm to low rpm transition PEM sequence: 1. Disengage clutch. Power disengagement involves retracting actuators simultaneously. 2. Power motor to maintain controller and ring gear somewhere between 0 rpm and an upper limit. The upper limit depends on the dimensions of the system and the sensitivity that this upper limit might have on the rotors performance. 3. Increase controller rpm to slow planetary carrier according to a predetermined optimum speed change rate (approximately 5-10 rpm per second main rotor equivalent) 4. The PEM will now seek to keep the rotor rpm stable by controlling the motor torque and speed. Low rpm to high rpm transition PEM sequence: 1. Slow the ring gear at a predetermined optimum slowing rate (approximately 5-10 rpm per second main rotor equivalent) by powering the motor. 2. Once ring gear has been brought to a slow speed of somewhere between 0 rpm and an upper limit, the clutches can be engaged. This upper limit depends on the amount of frictional force that can be developed by the clutches without considerable wear. 3. Power to the clutch actuators is discontinued and the clutches engage. The ring gear is now held stationary and the drive system returns to normal operating mode
High RPM: 650 fps main rotor tip speed Low RPM: 420 fps main rotor tip speed
Figure 13. Power Electronics Module architecture
CONCLUSION High speed flight has always been one of the greatest challenges in the Rotorcraft industry and there is an increased interest in maximizing forward speed without much compromise on weight and hover performance. This paper offered a promising design solution to slow the main rotor using the transmission. The new design incorporates the well understood planetary reduction system but achieves its speed reduction by allowing the ring gear to rotate from stationary to a predetermined speed. Through a controller motor and gear, the ring gear’s speed can be maintained, resulting in up to 50% reduction in rotor tip speed while allowing the engines to operate at their design range. This transmission design was adopted and tested on a theoretical helicopter for the AHS student design competition. The design transmission was optimized with a genetic algorithm to ensure the lightest weight system. The resulting rotorcraft achieved a top speed of 250 knots. This was 40 knots improvement over the single speed maximum. This speed increase was mostly attributable to the delay in drag divergence of the advancing blade through the speed reduction. This helicopter retained the low speed maneuverability and hover capability of a conventional rotorcraft and was free of instabilities through its flight regime. Additional work is needed: To accurately quantify the torque required for the controller mechanism to produce the desired control. To explore other control mechanisms such as traction based internally driven system. Perform controller simulations to test and optimize the Power Electronics Module control system. Test the entire design in a virtual environment before it could move on to actual physical tests.
APPENDIX Genetic Algorithm Overview A genetic algorithm is based off the theory of evolution. Due to its random nature, it cannot guarantee an optimum but has an increased chance of finding a global optimum in the presence of many local optimums as would be expected in this transmission optimization. The program builds a population of transmissions then selects the ―most fit‖
transmissions based on a predetermined criteria or overall evaluation criteria function. In this case the overall criteria were weight with all stress requirements being meet. The fittest parents are then ―mated‖ so that their combination produces a new population of transmissions that are better. This continues until there is no improvement in several successive generations. To increase the search ability, several offspring are also allowed to mutate or differ from either parent. Each design is represented by a binary string. At the beginning of a genetic algorithm, the design space is discritized into discrete steps. Due to the nature of the binary string, the genetic algorithm works only on discrete values. Continuous variables must be discritized into small steps. All variables are then converted into binary strings and those strings are then assembled end to end to form one long binary sting that represents that design. Initially, a population of points is randomly generated. The overall string length is determined by converting all variables and there ranges into binary strings and piecing these end to end as specified above. A population of points, typically 50-100 or so, are then randomly generated by randomly selecting 0 or 1 to fill each location in the sting. The stings are then converted back into decimal numbers and these stings and decimal equivalents are then passed onto the selection stage. There are many ways to determine which designs are promoted into the parent population to mate and create the next child population. The method used in this paper was a multiple elitist tournament selection method. This type of method helps the program converge faster which was useful in this paper since the sizing code, written in Matlab, took about 30-60 seconds a run to size each transmission for a given set of parameters. For this method, two strings were randomly selected and their translated decimal equivalents were run through the sizing program. Their overall function values (transmission weight in this case) were then compared. If the overall function values were among the top 1/3 of all compared thus far they were automatically promoted to the parent population. This is the multiple elitist part of the selection method. If they were not part of the top 1/3 then their two function values were compared and the one with the best function value (lowest weight) was then promoted to the parent population. This was continued until the parent population was the same size as the pervious population. The multiple elitist part of the strategy automatically promoted the top third of the designs into the parent population. This ensured the best designs were used in the parent population and helped to make the optimization more exploitive and converge faster. Other methods, such as the roulette method, do not guarantee that all of the
best designs are promoted. The main disadvantage of this type of method is that it tends to be exploitive rather than explorative. When there is a greater chance for lower ranking designs to be used in the parent population the algorithm naturally bounces around the design space more but converges slower. This disadvantage was offset by the tournament selection part of the method. Two designs were randomly selected, their parameters run, and there function values compared. In other methods, such as again the roulette method, the chance of a point being selected is directly proportional to its objective function value. This type of algorithm will quickly zero in on an optimum which, if done too quickly without adequate exploration, can many times be a local rather than global optimum. This method also has the added advantage of speed in that is a point is randomly chosen more than once, the algorithm can recognize this and not run the sizing program for it again. If this occurs, then all points in the population will not be fed into the sizing program increasing speed but sacrificing exploration. Once the parent population is chosen they are mated to create an offspring child generation. The parent population is grouped into sets of two randomly. Crossover takes place with a certain specified probability. If chosen, that set is mated and undergoes a two-point cross over. In this cross over method, the size of the bit to be crossed over is randomly selected. Thus, there is a chance that if the two numbers were sequential then there was a uniform cross over (all bits swapped). This is illustrated below for two points A and B. In this example Strsize is the total number of bits in the string.
Two Point Cross Over Example
Once this new generation is created from the parents the individual members are tested to see if they will undergo a random mutation. This mutation would swap a bit in a single position. For each member of this new population a random number was chosen between 0 and 1. If this number was less than the probability of mutation another random number was chosen, scaled, and rounded as above to the length of the string. The bit corresponding to this position was then changed from a zero to a one or vice-versa. Once this mutation is completed for all members, that population is then re-designated as the new parent population. The process of crossover and mutation is now performed on the new population. This is repeated until there is no change in the
objective function of the best design in a child population for a predetermined number of generations. Once this occurs the optimization is considered to have converged.
ACKNOWLEDGEMENTS The authors of this paper would like to thank all members of the Georgia Tech graduate design team for the 2009 American Helicopter Society Annual design completion. The design was truly a team effort. REFERENCES [1]
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