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Aerodynamic Analysis of Paper Gliders

Winter 2004


1.0 Introduction

The objective of this project is to compare the flight characteristics of gliders and understand the parameters that affect their performance.  Two gliders were used with slight variations in terms of airfoil and overall glider configuration. The configuration included such parameters as surface area, wing span, average chord, aspect ratio, root chord, taper ratio, mean aerodynamic chord, dihedral angle, and sweep angle.

The two gliders from a Whitewings® hobby glider kit were experimentally flown from a fixed point and the corresponding range was recorded. Qualitative observations were also made pertaining to how the gliders behaved during flight from launch to land points. By determining the glider that delivered greatest range, further analysis was performed on the gliders to understand why the measured range was different between gliders. For example, one of the planes may display better stability due to a smaller sweep-back angle. Other parameters of interest that were calculated included lift and drag components, lift-to-drag ratios, maximum lift-to-drag ratios, and the corresponding lift and drag components.

Further discussion on the experimental results included possible sources of error and limitations of the experiment. Recommendations are provided to detail how the experiment can be optimized to generate more representative data, as well as how other parameters can be investigated to develop a better understanding between the glider’s configuration geometry and range.


2.0 Procedure

Using the glider kit, two gliders were selected.  The criteria for selecting the gliders were based on selecting similar gliders with slight modifications, in order to conduct a comparable test.  The two gliders differed from each other in terms of different wing and tail shape, dihedral angles, and winglets.

1.1      Construction

The model gliders were pre-designed and constructed as per instructions given in Whitewings® instruction manual.  The completed gliders are as shown in figure 1. 

Glider 1

Glider 2

Figure 1: Glider 1 (elliptical) and glider 2 (trapezoidal)

1.2      Static Measurements

Various measurements such as surface area, weight, center of gravity, dihedral angle, chord length, sweep angle, wing span, and aerodynamic chord were calculated.

1.3      Test Flight

Before flying, each glider was inspected thoroughly to make sure that there were unwanted warps, bends or twists, and that the glider was evenly balanced.  Furthermore, the gliders were tested and continuously adjusted to optimize flight characteristics.  Adjustments were made based on whether the glider path curves left or right, and whether the nose goes up or down.  If the glider curved left or right during flight, the vertical stabilizer was adjusted accordingly, or the trailing edges of the wings were bent to simulate the effects of ailerons.  For the glider with the dihedral, the dihedral angle was slightly angled up or down to resist rolling and to produce a straighter flight.  The second factor that was tested was determining whether the glider was capable of trimmed flight.  If the nose of the glider pulls up too much, the glider will eventually stall.  To correct this, the trailing edges of the horizontal stabilizers were bent downwards.  If the glider started diving after launch, the horizontal stabilizers were bent upwards to compensate for the negative pitch. 

1.4      Testing and Dynamic Measurements

For the purpose of the experiment, the gliders were launched with a consistent force at an angle of attack of zero.   The gliders were launched at a height of six feet, as shown in Figure 2.  The launch force to the gliders was applied at the center of gravity.  The distance travelled by each glider and the flight time were recorded.  The distance was measured from the launch point to the first point the glider hit the ground.  Similarly, the time was measured from the initial launch to the time the glider first hit the ground.  This was used to calculate the speed of the glider as well as other aerodynamic parameters.


Figure  2: Launch height

 The planes were launched at positive, zero, and negative angles of attack.  The characteristics of these flights were also altered by modifying airfoil and horizontal stabilizers to find the best trimmed flight for each glider. 

3.0 Aerodynamic Parameters

The magnitude of forces, pressure, and moments that act on the glider depend on the combined effects of many different variables. Variation of these parameters will indirectly affect flight characteristics such as lift, range, drag, and speed. The parameters that have a direct effect on range are listed below.

·         Configuration geometry

·         Angle of attack

·         Glider size or model scale

·         Free-stream velocity

·         Density of the undisturbed air

·         Reynolds number

·         Air speed

 The parameters listed vary between the two gliders and must be taken into consideration when attempting to achieve optimal flight characteristics. In practice, flow phenomena such as turbulence, plane defects such as those caused by repeated crash landing, and uneven weight distribution (due to the glue used in construction) all limit the accuracy and range of flight.  The configuration geometry will be investigated in detail to observe how changing its associated parameters will affect the glider range. Lift and drag components will then be calculated to understand possible correlations between the varying parameters and range. It is to be noted that the gliders used in the experiment are approximated as flat plates when calculating Reynolds number. In reality, the paper airfoil can be assumed to be a thin airfoil and constant.  Therefore, the actual camber will not be discussed in much detail besides the fact that it generates lift due to pressure difference between the top and bottom surface.  The following sections explain the configuration geometry and other parameters for the tail and wing.

1.1      Measured Parameter

There are various parameters that can be directly measured off the gliders that related to the configuration geometry.  The different parameters that measured were measured are explained and detailed below. 

1.1.1       Wing Shape

Glider 1 has an elliptical wing and tail shape.  Glider 2 has a slightly tapered trapezoidal wing which includes a dihedral angle in the distal half of the wing span. 

1.1.2       Surface area

The surface area, S, is the plan surface area of the wing or tail. It is the major source of drag, since the drag on the fuselage is ignored for this project.  For the gliders that were tested, the surface areas were determined by importing the section schematics into AutoCAD and then determining the area from the commands available on the software. 

1.1.3       Wing and Tail span

The wing and tail span, b, is measured tip to tip of the wing or tail under study. The span can be a wide range values as it also depends on the sweep angle. For the tested gliders, the wing span was determined on AutoCAD by a similar technique as determining the surface area.

1.1.4       Root and Tip Chord

The root chord, cr, represents the chord at the wing centreline, and the tip chord, ct, represents the chord at the wing (or tail) tip. The ratio of the tip chord to root chord is known as the taper ratio, λ, defined by the following expression.


The taper ratio affects the lift distribution and the structural weight of the wing (or tail). A trapezoidal wing (or tail) has a taper ratio of 1.0 while the pointed tip delta wing (or tail) or elliptical wing (or tail) has a taper ratio of 0.0. These aspects will be taken into consideration when measuring the root and tip chord on the wings and tails of the test gliders.

1.1.5       Average Chord

The average chord, c, essentially represents the geometric average. It is to be noted that the product between the span and the average chord is the wing (or tail) area (b X c = S). The average chord was physically drawn on the wing and tail by joining the points of half lengths between the trailing and leading edge.

 1.1.6       Center of Gravity

By altering the gliders’ center of gravity with relations to the main airfoil, the stability of the glider can be greatly affected.  For example, changing the center of gravity from quarter chord to half chord will make the glider less stable for a given plane.  One way to counter this effect is to adjust the center of gravity by adding weights.  It is obvious that additional weight will reduce the glider performance.  Another way is to alter the shape and size of the rear stabilizer to achieve trimmed flight.  

1.1.7       Dihedral Angle

The dihedral angle is the angle between a horizontal plane containing the root chord and a plane midway between the upper and lower surfaces of the wing. If the wing lies below the horizontal plane, it is termed the anhedral angle. The dihedral angle affects the lateral stability of the airplane. The test gliders were kept at a constant dihedral angle for the wings and tails since it was a sensitive property to adjust on the glider. The dihedral angle can be positive or negative in value.  A positive dihedral has wing tip higher in elevation than the wing root.  This feature is added for better roll stability.  Dihedral is most often applied to planes that value stability over manoeuvrability. For example, commercial airliners have dihedral angles added to the wings.  

1.1.8       Sweep Angle

The sweep angle, Λ, is measured as the angle between the line of 25% chord and perpendicular to the root chord. Sweep angles of the leading or trailing edge are often the parameters that are presented. The sweep angle has considerable effect on the maximum lift and stall characteristics.

The following figure illustrates the parameters discussed on various wing configurations.

Figure 3: Sweep angle for various wing configurations


1.2      Calculated Parameters

Using the measured characteristics, various parameters important to flight can be determined analytically.  These are explained and detailed below.

1.2.1       Aspect Ratio

The aspect ratio, AR, is defined as the ratio of the span and the average chord. For a trapezoidal wing (or tail), the aspect ratio is given by the following expression:


For a nonrectangular wing (or tail), the aspect ratio is given by the following expression:


The aspect ratio is essentially represents the “fineness ratio” of the wing (or tail) and is very useful in determining the aerodynamic characteristics pertaining to induced drag.

High aspect ratios represent wings (or tails) that typically have long spans or thin chords while low aspect ratios represent short spans or thick chords. Gliders typically have large aspect ratios while planes such as aerobatic or military planes will have small aspect ratios. One major advantage to using large aspect ratios for glider planes is that large aspect ratios will produce lift without being penalized by the drag forces. Smaller aspect ratio typically sees more induced drag as “induced drag depends inversely on the aspect ratio.” [1] 

1.2.2       Mean Aerodynamic Chord

The mean aerodynamic chord (mac) is used with surface area to non-dimensionalize the pitching moments. It represents an average chord when multiplied by the product of the average section moment coefficient, the dynamic pressure, and the wing area, the moment of the entire wing is produced. The mean aerodynamic chord can also be written by the following expression.




To identify the mean aerodynamic chord on the gliders’ wings (and tail), the following figure illustrates the geometrical approach that was used.  Appendix B shows the calculated parameters.

Figure 4: Geometrical approach for determining mac and centre of gravity 

1.2.3        Glide Ratio

The glide ratio is a macroscopic measure of the overall glider performance.  For a given launch height, a longer flight distance will provide a smaller glide ratio as seen in figure 5. As mentioned before, this ratio is the same as the lift-to-drag ratio, or L/D. Therefore, it can be said that designing for the maximum L/D ratio will result in the best glider performance.

Figure  5: Trigonometric representation of glider decent path


1.2.4       Lift and Drag

Lift is affected primarily by speed of the flow over the airfoil as well as camber, angle of attack, and angle of incidence.  Drag, on the other hand, acts in the direction opposite to the glider’s motion.  The components of drag can be stated by the following expression.


 Skin friction and form friction combine to form parasite drag. The effects of induced drag are captured by the second term. Since the value of K can only be determined by experimental means, the efficiency factor is assumed to be 1 and 0.60 for glider 1 and glider 2, respectively.  The definition of the efficiency factor is provided below.


Induced drag is caused by vortices shedding off the tips of the wings. The wing tip vortices produce a down wash of air behind the wing which is very strong near the wing tips and decreases toward the wing root. “The local angle of attack of the wing is increased by the induced flow of the down wash, giving an additional, downstream-facing, component to the aerodynamic force acting over the entire wing.” [1] The calculation of the induced drag coefficient is shown below.  The induced drag coefficient varies with the square of the lift coefficient.  Hence, induced drag is also called “drag due to lift”.


Induced drag can be reduced in two ways: high aspect ratio and elliptical wings.  The reason that high aspect ratio wings is effective in reducing induced drag is that the longer the wingspan the less the effect of the vortex downwash has on the entire wing surface.  Lifting line theory shows that the optimum (lowest) induced drag occurs for an elliptic distribution of lift from tip to tip. The efficiency factor e is equal to 1.0 for an elliptic distribution and is some value less than 1.0 for any other lift distribution [2].  As mentioned in class, the British WWII Spitfire is an excellent example of the performance offered by the elliptical design. 

By knowing the angle of decent, glider weight, and the dynamic pressure, the lift and drag components can be calculated by the following expressions.





It is known that for steady level unaccelerated flight (SLUF), the moment about the glider’s centre of gravity is equal to zero. The following equation can be used to determine the lift force on the tail and wing.





The values dW and dH correspond to the distance between the wing and tail aerodynamic centre to the centre of gravity which was determined as the point 25% of the mean aerodynamic chord. By determining dw and dH and assuming W=L for the SLUF condition, the lift and drag forces can be determined by calculating the lift and drag coefficient by the dynamic pressure.  

The lift-to-drag ratios can be calculated by dividing the total drag by total lift, or by the following simplified expression.


1.2.5       Maximum Lift-to-Drag Ratio

The configuration and the application of an airplane is closely related to the lift-to-drag ratio. Many important items of an airplane performance are obtained in flight at (L/D)max.

The maximum lift-to-drag ratios as well as the lift and drag components associated with the maximum lift-to-drag ratios are stated below.





4.0 Results and Observations

Table 3 summarize the results of the measured and calculated parameters described previously.  Appendix C show the data recorded from the experiment.



Glider 1

Glider 2






Weight (g)




Range (m)




Average Flight Time (s)




Height (m)




Angle of Decent (radians)






Area (cm2)






Wingspan (cm)






Average Chord (cm)






Aspect Ratio






Root Chord (cm)






Tip Chord (cm)






Taper Ratio






Dihedral Angle ( ° )






Sweep Angle






Mean Aerodynamic Centre (cm)






Centre of Gravity (cm) from nose

c of g



Efficiency factor












Centre of Gravity to Wing






Centre of Gravity to Horizontal Stabilizer






Induced Drag Coefficient






Induced Drag 






Parasite Drag Coefficient






Parasite Drag 






Total Drag Force (N)




Lift Coefficient






Lift Force (N)






Total Lift Force(N)






Drag Coefficient






Drag Force (N)






Moment Coefficient












Lift-to-Drag Ratio






Table 1: Results and observations

The following figures illustrate the maximum and mean flight range for both gliders at three different angles of attack.  The longest flight was achieved with zero angle of attack followed by -10 degrees, followed by +10 degrees of attack.  This is true for both the maximum and mean flight distance with the exception of maximum flight distance at 10 degrees angle of attack.


Figure  SEQ Figure \* ARABIC 6: Maximum and mean flight range versus various angle of attack of test gliders


One other trend is apparent when comparing the two gliders.  Glider 1 was capable of achieving longer flight when looking at the maximum flight range.  However, glider 2 was superior when comparing the mean flight range.  Overall, it can be concluded that glider 2 is more consistent but glider 1 is capable of reaching longer distance. 


Glider 1 and glider 2 had similar flight characteristics; however, comparing the results of the best straight line distance, glider 1 flew further than glider 2.  Both gliders had a tendency to deviate off the straight line path.  Modification to the airfoils to simulate ailerons minimized the deviations, but were only effective short-term until the glider collided with a wall. 


When launched with zero angle of attack, neither of the gliders experienced trim flight without constant adjustment.  The horizontal stabilizers were altered to simulate elevators as mentioned earlier in the test flight procedures.  As mentioned before, all measurements and observations were made only for trimmed flight.  This is the simplest and most macroscopic measure of optimized flight.  In the trimmed condition, the moment about the plane’s center of gravity is zero.  When the plane oscillates, it indicates that there is a pitching moment acting about the center of gravity which reduces the efficiency of the plane and reduces flight range.  Symptoms of untrimmed flight can be noticeable right after the launch.  If the plane is launched at zero degrees to the horizontal, it should maintain this angle right after launch.  Untrimmed flight will exhibit a tendency for the plane to dive or “pull up”. When the plane pulls up, it gains altitude until stalling.  At this instant, it will dive to regain speed until the moment acts to pull up the plane again.  This cycle of energy transformation between the kinetic and potential continues until the plane eventually lands.  The same energy transformation occurs for a plane that initially dives. 

 5.0 Discussion

1.1      Flight Analysis

Plane 1 (elliptical):

The advantage of the elliptical wings, due to higher efficiency, should yield longer flight distance than plane 2 due to the lower induced drag with everything else being equal.  Experimental results were not conclusive in determining a definite advantage in distance.  The measured linear flight range of the two planes was quite similar.  In fact, the differences between flights were much more significant than any differences between the two planes.  Plane 1 seems to be more sensitive to turbulence and veered off course more so than plane 2, even with adjustments between flights.  The actual distance of the flight, had it not veered off course, is difficult to measure.  However, for the few flights which followed a straight trajectory, plane 1 was superior to plane 2.  Therefore, it is reasonable to say that the experimental results agree with the theoretical prediction.  That is, longer flight is produced as a result of lower induced drag.  This is, of course, assuming that the difference in lift is minimal as well as the other parameters being rather insignificant.         

Plane 2 (trapezoidal with dihedral):

The flight of plane 2-with trapezoidal airfoil and dihedral-proved more stable and is more resistant to rolling.  The reason for the more consistent flight of plane 2 has to do with the dihedral.  The deflection of the distal half of the airfoils upward is analogous to a positive dihedral angle.  This dihedral contributes to better stability in crosswind (in the form of hallway turbulence) and more consistent and straighter trajectory.  Although the tests were carried out in hallways and large lecture halls with little turbulent airflow, the effect of such flow were very significant on the paper gliders.    Experimental results show that during most of the flights, plane 2 was able to maintain a straight trajectory, at least more so than plane 1.  However, by comparing the longest flight distance it was shown that plane 2 was inferior to plane 1. 

Zero angle of attack produced the longest distance when the glider was trimmed properly.  This is not at all surprising.  The reason that the +10 degree was the worst case scenario can be explained in terms of stall behaviour.  After stalling, the gliders dived sharply, and lost altitude very quickly without gaining any distance.  For the -10 degree angle of attack, the plane dived gradually and generated distance while losing altitude.

Based on the results, both gliders have a similar surface area for the wing and tail.  However, glider 1 has a lower drag force and a higher lift force.  This accounts for the greater range when compared to glider 2. The longer distance can also be confirmed by the higher wing aspect ratio of 5.78 for glider 1 compared with 1.8 for glider 2.  The results conform to theory that indicates higher aspect ratios result in higher rate of lift with less drag [2].  In general, the larger the aspect ratio results in larger lift/drag ratio. This was observed in Table 2.   

1.2      Sources of Error

1.2.1       Glider Construction

The glider was built out of thin cardboard and glued together based on the instructions provided.  It is possible that an imperfect construction could lead to less optimized flight performance.  For instance more glue might have been used on one side of the wing, which will cause an imbalance.  This is a possible explanation to why the gliders tend to veer off-course due to a rolling moment.  Furthermore, the dihedral angle was created by bending the cardboard.  Although care was taken during construction, the angles can be inaccurate. 

1.2.2       Glider Launching

The glider was launched by hand, which can give rise to many possible errors.  The force applied to the gliders was probably inconsistent.  This will result in flawed distance and duration values.  It is also possible that due to the numerous test flights that the force was not applied at the center of gravity during every launch.  This causes the glider to have a positive pitching moment.  Furthermore, it is unlikely that the various angles of attack for was consistent because of the hand launch.  Different angles of attack can have great effect on the distance and the duration of flight.

1.2.3       Glider Landing

After initial construction, changes were made to the wing, fin, and tail to produce a smooth straight flight.  However, during the experimentation the gliders hit the walls and other obstacles quite frequently, and had several hard landings.  This caused slight geometric changes, such as changes to the dihedral and winglet angles.  These changes undoubtedly caused variations to the glider’s flight characteristics.

1.2.4       Measurement Errors

The flight paths of the gliders were not straight. This caused a curvilinear trajectory on the horizontal plane. This made measurement difficult since tracing the path after landing was inaccurate.  The other measurement errors are based on human inconsistency and error, such as timing errors.

 1.3      Limitations

Due to the availability of limited resources for this project there were many limitations.

  • The model kit used prevented the gliders to be modified greatly.  For instance, it was not possible to move the location of the wings and change the center of gravity.
  • The model kit used did not have any rudders, ailerons, or elevators.  This prevented controlling the gliders’ flight path precisely.
  • Without constructing some sort of mechanical launcher, the initial launch force and the angle of attack could not be measured.
  • The lack of large indoor space without obstacles prevented the gliders to be launched with a larger force.
  • The data collected is not statistically useful.  Changes have to be made between each flight in order to assess the effect.


6.0 Conclusions

The objective of the project was to compare the flight characteristics of gliders and understand the parameters that affect their performance, particularly range.  Two gliders were used with slight variations in terms of configuration geometry. The configuration geometry included such parameters as surface area, wing span, average chord, aspect ratio, root chord, taper ratio, mean aerodynamic chord, dihedral angle, and sweep angle.  Based on these parameters and experimental results, plane 1 is capable of reaching greater flight distance.  However, under most circumstances, plane 1 is rather susceptible to turbulence and produces less than optimal flight characteristic.  A good way of looking at this is from the point of view of a glider competition.  If the best flight distance is desired, then glider 1 is the glider of choice.  If consistent results are paramount, then glider 2 will offer good performance with high consistency.    

7.0 Recommendations

The gliders that were tested were launched from a fixed point by simply throwing at a fixed angle of attack.  In order to reduce the error involved in launching, it is recommended that more trials be conducted for each glider produce a better statistically representation of the flights.  Another suggestion for improving the launch is by constructing a glider launch pad. The launch pad will consist of a flat, smooth, variable angle stand that would have a groove machined down the centre.  An elastic will be attached to “slingshot” the glider into flight.  This will deliver a takeoff with consistent force and angle of attack.

 This experiment gave us an idea of how certain parameters affect flight,  In order to get a better understanding of the effects of the other variable such as center of gravity, dihedral angle, geometric twist, angle of attack, it is recommended that the experiment be redesigned.  The redesigned experiment would involve changing one parameter at a time and observing the flight characteristics.

8.0 References

1.         NASA Glenn website: http://www.grc.nasa.gov/Doc/educatn.htm

2.         John J. Bertin, Aerodynamics for Engineers, 4th Edition, pp. 156-192, Prentice Hall, New Jersey, 2002.

3.         Yasuaki Ninomiya, Whitewings: Introduction to Paper Plane Design, pp. 24-35, AG Industries Incorporated, Redmond, 1980.

4.         Airfoils and Airflow, http://www.av8n.com/how/htm/airfoils.html, See How it Flies: A new spin on the perceptions, procedures, and principles of flight, Date Accessed: March 5-25, 2004