Vahana Configuration Trade Study — Part II
In Part I of the Vahana Configuration Trade Study we outlined the two configurations selected for in-depth study: an electric helicopter and an electric eight fan tilt-wing. Additionally, we described a representative mission and discussed some of our technology assumptions.
In Part II, we’ll dive into details of the multidisciplinary design optimization (MDO) problem used to size both configurations and present our trade study results.
In keeping with our belief at Vahana that sharing information and collaboration makes ideas take off, and for those interested in an in-depth look at the MDO problem, we’re also sharing the Matlab® source code used to generate the results presented here on GitHub. We hope that readers take this opportunity to learn about aerodynamics or structural analyses and test their own assumptions.
MDO Sizing Problem
As in all optimization problems, we have to define three key elements:
- The objective function we are trying to optimize.
- The design variables we are allowed to vary.
- The constraints that the design must satisfy.
The objective and constraints are formulated as functions of the design variables, which we refer to as models and will describe briefly in the following sections. A depiction of the data flow between the various models and functions is shown below.
Traditionally, aircraft designers have used many different objective (cost) functions. They can be as simple as minimizing the maximum takeoff weight, or as complicated as forecasting the expected Net Present Value (eNPV) over a full product lifecycle. We decided on an intermediate objective: direct operating cost (DOC). DOC captures most of the economics associated with the performance and operations of an aircraft. There are clearly other objectives that we care about for VTOL aircraft such as the noise footprint and safety / reliability metrics, but these were not quantified at this stage.
We developed our DOC model loosely based on the Air Transport Association (ATA) DOC model for transport aircraft. Costs are broken down into the following components:
- Structural materials, tooling, and manufacturing
- Component costs (battery, motors, servos, avionics)
- Parachute recovery system (for tilt-wing only)
- Facility rental cost for storage and operations
- Electricity cost ($0.12 per kW-hr)
- Maintenance (labor and part replacement)
Inputs to the model include the design variables described below and many assumed parameters such as the vehicle lifetime and labor costs. As with the ATA model, ours captures costs that are fixed per year, costs dependent on the number of flights, and costs dependent on the number of flight hours. The result is a cost model that captures the major contributions to vehicle operating cost and properly represents the utility of speed.
The following high-level design variables were used:
- Main rotor radius (helicopter), Fan radius (tilt-wing)
- Design cruise speed
- Maximum takeoff weight
- Battery weight
- Motor weight
For the tilt-wing configuration the wingspan is assumed to be a linear function of the fan radius. For the helicopter, the tail boom length and tail rotor radius are functions of the main rotor radius.
We required each vehicle to satisfy a handful of critical and explicit constraints:
- The energy required to complete the reserve mission must be less than the battery energy available
- The maximum power required from an individual motor must be less than the motor power available
- The calculated takeoff weight must be less than the maximum takeoff weight design variable
- The helicopter rotor must have enough kinetic energy for the flare at touchdown after an auto-rotation
It should be noted that the first three constraints above are compatibility constraints. These are used instead of nested internal iteration loops, which you see in most aircraft sizing codes. The numerical optimizer is able to efficiently satisfy these constraints while minimizing the objective function. Also, many additional constraints were codified into the models outlined below.
Aerodynamic Performance Model
The size and performance of both the tilt-wing fans and helicopter rotor are strongly driven by a maximum blade tip Mach number constraint of 0.65 that we specified for noise reasons. The helicopter rotor RPM is set by this constraint in cruise, while for the tilt-wing it is applied at the 1.7 times hover thrust condition (to allow maneuvering margin with a failed motor).
For both configurations, the hover performance estimates were based on blade element momentum theory with corrections for tip losses and assumed values for the blade profile drag coefficient and solidity.
For the helicopter, the cruise performance is also based on blade element momentum theory. The wing area for the tilt wing is set by the specified cruise configuration stall speed of 35 m/s. The tilt-wing cruise performance is based on a traditional quadratic drag polar with a span efficiency factor of 1.3, which is achievable with a non-planar tandem wing configuration. A conservative fuselage and excrescence drag area is assumed to be the same for both configurations.
The transitions from hover to cruise flight and back are not explicitly modeled, and no range credit is applied for these segments. We also made the conservative assumption that the power required during transition is the same as the power required in a stationary hover.
For both vehicles we also specified maximum power conditions that size the electric motors. For the tilt-wing this is the 1.7 times the hover thrust. For the helicopter, a 10% thrust margin is assumed at hover, and an additional 15% power margin is added to account for the tail rotor at its maximum control authority.
The various component weights are estimated based on Federal Aviation Regulations (FAR) derived load cases, the geometry determined by the design variables, and historical trends for a limited number of components. A common payload weight of 250 lb (113 kg) was assumed for both vehicles to represent a single passenger with a carry-on bag.
The wing and canard weights for the tilt-wing assume a bonded structure consisting of carbon fiber spars and skins with aluminum ribs. Lift, drag, torsion, and thrust loads are considered during sizing. The rotor and fan blade weights assume a cored composite structure to take the centrifugal, flap, lag, and torsional loads and an aluminum fitting to connect to the hub.
The fuselage weight assumes a carbon fiber structure sized by both the aerodynamic and landing loads. The landing gear weight for both configurations was assumed to be 2% of the maximum takeoff weight based on historical trends for helicopter skids. One major component unique to electric vehicles is the power distribution wire weight. We first estimated the power distribution wire weight based on the total power going to each motor and the motor locations in the vehicle. We also include estimates of the weight of low voltage power and communication wiring for avionics and sensor systems.
For the helicopter we assume a gearbox with a power density of 6.3 kW/kg to allow for redundant power input from multiple electric motors and to achieve the desired rotor RPM.
Both configurations assume a 15 kg allocation for avionics components, and a 15 kg crash rated seat. The electrical actuators are assumed to weight 0.65 kg each, with eight for the helicopter (redundant collective, cyclic, and tail rotor) and 12 for the tilt-wing (four control surfaces and eight variable pitch). Additionally, the tilt-wing has two actuators that each weigh 4 kg to control the wing tilt angles. Finally, we added an additional 10% margin to both configurations to account for fittings and miscellaneous hardware.
Trade Study Results
We ran the MDO problem described above for a series of design ranges between 10 and 200 km. The first result we’ll examine shows the calculated DOC per km for each of the two configurations optimized at each design range. Keep in mind that each dot here represents a unique vehicle optimized for a given range. The size of the dot represents the takeoff mass of that vehicle.
First, you’ll note that at short ranges the two configurations perform similarly, with the helicopter having a small advantage. As the design range reaches about 70 km the DOC curves cross. As we reach 100 km, the helicopter struggles to close and only the tilt-wing can achieve the design range (while also maintaining a low DOC). You can see why this is the case by examining the mass breakdown of the two configurations as a function of range, below.
The green portion of the bars represent the battery mass, and as expected the battery weight starts to dominate both configurations as the design range increases. The faster rate of increase in battery mass with range for the helicopter is due to the lower cruise lift-to-drag ratio compared to the tilt-wing.
Most transport aircraft have a maximum fuel weight that is roughly 1/3 of the maximum takeoff weight. If we look at the designs where the batteries are about 1/3 of the takeoff mass, the electric helicopter has a range of about 20 km and a DOC of $1.63/km, while the electric tilt-wing has a range of about 50 km and a DOC of $1.24/km. Next, we take a closer look at the breakdown of DOC for both configurations.
Part replacement is the largest component of DOC at most ranges, which is in turn dominated by the battery replacement cost. Here we’ve assumed 2000 flight cycles before the battery pack needs replacement. This sensitivity indicates that high cycle life pack designs that treat the batteries well will be important to minimizing DOC.
As with most electric vehicles, the energy cost is a small component of the overall DOC at about $0.10/km. For short ranges the labor cost per km is high due to the assumed pre-flight inspections and maintenance that is required for each flight.
By way of comparison, it’s worth noting that the minimum DOC is predicted to be about 1.5x that of a Tesla model S or about 4x that of a Toyota Camry. The primary advantage of VTOL aircraft comes from an average point-to-point speed which is ~2x faster than a car under most conditions.
We also looked at how some traditional aircraft and helicopter performance metrics vary with the design range for each configuration. The next figure shows the variations in hover power, cruise power, disk loading, and power loading with design range.
As expected the cruise power of tilt-wing is lower due to its larger lift-to-drag ratio. The disk loading for the tilt-wing is higher, indicating a fundamental tradeoff in the two configurations between hover power and cruise power. The disk loading for both configurations is similar to those of many light helicopters. At short ranges the helicopter hover power is lower, however, the faster increase in helicopter weight with range causes the helicopter to require more hover power for ranges above 50 km, even though it maintains a lower disk loading.
A conceptual design sizing tool like this is useful for performing sensitivity analyses to identify the most important assumptions and design parameters. Below we present a couple of example sensitivity analyses. We encourage you to play with the source code on GitHub to test your own assumptions.
The first sensitivity analysis looks at the change in vehicle takeoff weight and DOC for battery pack specific energy densities between 150 W-hr/kg and 500 W-hr/kg for the tilt-wing configuration with a 50 km range.
The next sensitivity analysis looks at the effect of battery cycle life on DOC for the tilt-wing configuration with a 50 km range.
Lastly, we examined the sensitivity of the vehicle takeoff weight and DOC to the reserve segment loiter duration (i.e. how much longer the vehicle can fly after completing its maximum range mission, in case there is an emergency that keeps it from landing at the desired location/time) for the tilt-wing configuration with a 50 km range.
At low design ranges a helicopter is shown to be a compelling configuration, while at longer ranges the improved cruise aerodynamics make the tilt-wing configuration more compelling. The exact DOC values and crossover point are sensitive to many assumptions inherent in such a conceptual design tool.
While the exact performance specifications of Vahana have not been finalized, we believe that the electric tilt-wing configuration provides a DOC advantage and many other advantages such as reduced noise and enhanced safety for urban mobility.
- Geoffrey Bower