Copyright ©2016 by Paul Niquette. All rights reserved.
Level flight requires a balance of aerodynamic forces -- especially in pitch. Stability requires that vertical perturbations will be correctable by aerodynamic control forces. Moreover, in the event of an engine failure, the aircraft must not stall by default. That means the center of gravity for an aircraft is located forward of the center of lift as shown in the puzzle.The resulting nose-heaviness in level flight must be balanced either by a downward force provided by the elevator located in the empennage or by an upward force provided by a canard located near the nose of the aircraft. In the Conventional Configuration, the required forces obey the following equation: Lift = Weight + Elevator. For the Canard Configuration, the equation must be changed to read Lift = Weight - Canard.
Nota bene, elevator is a misleading term. It controls pitch. Same for the canard, by the way. Pitch determines the angle-of-attack between the wing and the relative wind. Thus either the elevator or the canard controls angle-of-attack. It is thrust that really controls the "elevating" of an aircraft, whether climbing, descending, or in level flight. Loss of thrust results in loss of altitude but must not result in an increase in angle-of-attack toward a stall condition. Thus, for safety, center of gravity must be located forward of center of lift.
Balancing forces, Elevator and Canard, are provided by air flowing over and under the respective control surfaces. The expression 'pitch authority' can be used to characterize the 'torques' on the fuselage resulting from those two forces. Torque is proportional to the product of the surface area of a control and its moment arm.Solvers are invited to use the sketch below, which depicts the respective moment arms for the aerodynamic forces in reference to the aircraft's center of gravity....
Sophisticated solvers might make the assumption that xWL is the same for the two configurations then calculate the balancing torques with respect to each aircraft's center of gravity for each configuration.
The ratio of the required balancing force on a canard compared to an elevator varies inversely with the length of their respective moment arms. Here are three interesting cases...
Case 1. xWE = xWC, Canard = Elevator
Case 2. xWE > xWC, Canard > Elevator
Case 3. xWE < xWC, Canard < Elevator
cases, there is
Case 2 would
seem to offer
for the canard
of the Tin-Can Mystery puzzle will recall that surface
area of an
as the square
of a linear dimension,
of the canard
Solvers who study the historical collection of canard aircraft listed in the puzzle will find configurations with a wide variety of moment arms and control surfaces.Let us conclude by analyzing the Boeing 767 canard configuration postulated in the puzzle. The sketch below applies the parameters defined above. The moment arm for the center of lift with respect to the center of gravity xWL is assumed to be the same for both configurations.
One might estimate that the moment arm for the Elevator xWE to be about twice the moment arm for the Canard xWC, which means the required balancing force Canard > Elevator by a factor of two, which corresponds to Case 2. There would be a penalty of about 83% of the weight of the elevator in the Canard Configuration, which is small compared to the Canard Advantage resulting from the reduction of requisite Lift, which solvers will find described in Wages of Flight.
Illustrated below is a postulated lengthening the fuselage for the Boeing 767, which exploits the Canard Advantage by applying the increase in useful load as accommodations for a larger number of passengers or space for freight...
Finally, the Canard Advantage includes a solution to the problem of ground effect under the elevator.
During its take-off roll, the pilot of a fully loaded transport aircraft must perform a rotation maneuver at a specified speed V-R, lifting the plane's nose to increase the angle-of-attack on the wings for lift-off. In the Conventional Configuration that means the elevator is being brought low to the ground such that pressure increases underneath tending to prevent further rotation. If the pilot delays rotation until after accelerating to a higher speed, the aircraft may not be able to take off before the end of the runway.