
Mr. Kevin Rose of Rose Aviation Service (Akron, NY) owns and flies a Fouga Magister, a 1960’s era jet trainer that was used by the French Air Force. As I am always on the lookout for the unusual to model, his Magister was the perfect candidate. Kevin graciously allowed free access to his Magister, which I crawled in, on, under and over every inch. Armed with a tape measure, I gathered the dimensions needed to draw a set of plans. The result was a 1/8 scale Magister, 56” wing span (12% Clark Y, 354 sq. in.) and weighing in at 48 oz. A Wemotech MF480 ducted fan driven by a MEGA 16/15/2 brushless motor (275 watts) powers the Magister.
My intentions were to handlaunch the Magister but two unsuccessful attempts persuaded me to try a bungee launch. Bungee launchers (think aircraft carrier catapults) are simple to make and use. Mine consists of a nylon leader, the bungee cord, a foot release, an anchor pin and a launch ramp. To use, just stretch the cord between the anchor and the foot release, place the Magister on the ramp, attach the leader to the Magister, step on the release and the Magister is airborne. Power is applied after the cord drops away.
Out of curiosity, I decided to analyze the bungee launch. Using a tape measure, a “fish” scale, my observations, an old Physics textbook, some assumptions and a flat out guess, I quantified the bungee launch.
A “rule of thumb” is the bungee tension should be about 5 times the weight of the aircraft. The cord I am using allowed a maximum stretch of 154 feet, resulting in 13 lbs. of tension, not quite 5 to 1 but close. Of this length, 10 feet is leader and 54 feet is the relaxed length of the bungee. So the actual stretch of the cord is 90 feet.
Where F is expressed in lbs., K is the spring constant expressed as lb./ft. and X is the spring stretch in feet. I measured the tension in 10foot increments and found the spring constant varied between .25 lbs./ft. at 10 feet of stretch to .144 lbs./ft. at 90 feet of stretch.
OBSERVATIONS
Tension is a vector quantity that resolves itself into horizontal and vertical components. The horizontal component is what pulls the Magister forward while the vertical component pulls downward and adds to the weight of the Magister.
Once released, the bungee cord is horizontal for only a fraction of a second. As the Magister accelerates it gains altitude and increases the cord angle with the ground. The horizontal component of the tension is reduced by the cosine of this angle.
The energy stored in the bungee cord is equal to ½* K*X^2 where K is the “spring constant” of the bungee and X is the stretch distance.
The kinetic energy of the Magister is equal to ½*M*V^2 where M is the plane’s mass and V is velocity in ft./sec.
I drew a scale launch trajectory that I divided into 10foot increments. Drawing lines that represent the cord at each point, I measured the length of the cord and the angle from the horizontal. From these measurements and F=K*X, I determined the total tension and both the horizontal and vertical components at each point along the trajectory.
Click to open these images.
I calculated the Magister’s mass by dividing its weight by the gravitational constant Gc (equal to 32.17); thus the 3 lb. Magister’s mass is .09325 “slugs” (the "slug" is the unit of mass in the English Engineering System and its measurement is lbs.*sec^2/ft. Its counterpart in the MKS (metric) system is the kilogram).
I then calculated the total energy stored in the bungee. Based upon my trajectory drawing I calculated a value of [½* K*X^2] for each 10foot increment. I then drew a graph using these values as the “Y” axis and the stretch length as the “X” axis. The calculated area under the curve was equal to 20571 lbs*ft^2. Dividing this by 90 feet yielded a stored energy of 228.6 ft*lb. It is important to note I have neglected all losses for this calculation.
Setting this value equal to ½*M*V^2, (kinetic energy of the Magister) yields a velocity of 70.0 ft/sec (47.7 MPH), assuming no aerodynamic drag.
I then plotted a second graph with total tension as the “Y” axis and stretch distance as the “X” axis. I determined the area under this curve to be equal to 735 ft*lbs. Dividing this by 90 feet yields an average tension of 8.2 lbs. Again, I have neglected all losses for this calculation
Solving F=M*A for A, using 8.2 lbs of force and a mass of 0.09325 slugs, the Magister’s acceleration is 87.6 ft/sec^2 or 2.7 G!!
From V=A*t, with V=70 ft/sec and A = 87.6 ft/sec^2, the elapsed time for the launch is 0.8 seconds, again, neglecting all losses.
Looking at the elapsed time from a displacement point of view and solving X = 1/2 A*t^2, with X = 90 feet and A = 87.6 ft/sec^2, the elapsed time for the launch is 1.43 seconds, again neglecting all losses.
I think it is safe to say the elapsed time is somewhere between these two values, again neglecting all losses.
Based on my assumed trajectory, the last interval of interest had a stretch distance of 64 feet, which is the length of the nylon leader plus the natural length of the bungee. Thus the cord tension is now zero and no further energy is imparted to the Magister. Since the tension in the cord is zero, there is nothing to keep the nylon leader attached to the launch hook. Due to drag on the bungee, the cord falls away. My trajectory drawing indicates the cord separates from the Magister about 50 feet before the anchor pin. The cord’s momentum and weight carries it forward and down, toward the anchor. This is borne out by my observations of the cord falling to the ground before reaching the anchor pin.
The maximum vertical component of cord tension was calculated at 44 feet down field, a bungee stretch distance of 111 ft. and 9 degrees from the horizontal. The cord tension added 0.8 lbs to the weight of the Magister for an apparent weight of 3.8 lbs. The vertical component decreased thereafter.
These calculations agree with my observations but without a means of measuring velocity and altitude, they cannot be confirmed. An accurate launch trajectory is crucial to the analysis but this requires very sophisticated video equipment. So my guess will have to suffice. In my fifteen (15)launches to date, the cord always fell to the ground before the anchor pin.
The effect of aerodynamic drag, which I assume to be 10%, may or may not be correct for the Magister. Further, my assumption of perfect energy transfer from the bungee to the Magister is not valid, the loss being somewhere between 25 and 35 percent. Considering 10% aerodynamic drag and and a 25% loss of energy from the bungee to the Magister, the Magister's velocity when the hook drops away might be closer to 63 ft/sec. If I assume a 2 G acceleration (25% loss of energy) the elapsed time for the launch is somewhere between 1.0 and 2.0 seconds. Not an unreasonable value when compared to my observation.
Changing the weight of the Magister, the launch hook location, the strength and/or length of the bungee cord will change the results. A more accurate scale would be helpful. Checking my scale against two gallons of water (16.6 lbs) leads me to believe the scale reads 10% high. !
The bungee launch is very effective when a hand launch can’t be achieved or your plane does not have landing gear. Prior to any first flight, it’s imperative to know where the CoG is located. If the plane is a kit, this is usually not an issue as the CoG has been established. But if the plane is a scratch built model, careful attention must be paid to this critical area.
The attachment of the launch hook to the fuselage requires some thought, as this is what must sustain the G forces of the launch. The hook has to be located ahead of the CoG and if it is going to come loose, it will do so during the bungee launch.
I have also made poweron launches. The EDF thrust adds to the cord tension and results in much higher altitude and velocity at time of release. While this is a subject for another investigation, I can report the cord still falls up field of the anchor.
I chose the Clark Y because of its high lift at low speed, making it an ideal wing for hand launching. If I were designing for a bungee launch I might have chosen a thinner airfoil that would result in less altitude but higher velocity.
A bungee launch gone badly is not a pretty sight. In about one second, the plane can accelerate to 60~70 ft/sec. At this speed, one can expect a lot of damage if the plane fails to get airborne and heads for the dirt instead.
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