maximization of flight duration of a free flight glider (doe)

MAXIMIZATIONOFFLIGHTDURATIONOFAFREEFLIGHTGLIDER

IEE 572 – DESIGN OF ENGINEERING EXPERIMENTS

Fall 2012

Term Project Report

Instructor: Dr. Douglas C. Montgomery

PROJECT TEAM

Arjun Gopal Radhamani 1205215995 (In Person)

Ramachandran Sundaram 1204102583 (Hybrid)

Sriram Arunachalam 1205035334 (In Person)

IEE572 Maximization of flight duration of a free flight glider

Page | 1 ArjunGopalRadhamani, RamachandranSundaram, SriramArunachalam

Contents Contents .......................................................................................................................................... 1

Figures ............................................................................................................................................ 3

Tables .............................................................................................................................................. 3

1.0 Introduction ..................................................................................................................................... 3

2.0 Pre-Experimental Planning ............................................................................................................. 3

2.1 Recognition of and statement of the problem ................................................................................. 3

2.2 Selection of Response Variable ........................................................................................................ 4

2.3 Choice of factors, levels, and range ................................................................................................. 4

2.3.1 Design Factors ............................................................................................................................... 4

2.3.2 Held Constant Factors .................................................................................................................. 6

2.3.3 Uncontrollable Factors ................................................................................................................. 6

3.0 Choice of Experimental Design ....................................................................................................... 5

3.1 Design ............................................................................................................................................... 6

3.2 Factor Type and Levels ................................................................................................................... 7

4.0 Experimental Procedure .................................................................................................................. 7

4.2 Experimental Run............................................................................................................................ 7

4.3 Design Matrix .................................................................................................................................. 7

5.0 Results and Statistical Analysis ....................................................................................................... 7

5.1 Actual response versus Predicted response ..................................................................................... 9

5.2 R-Square ........................................................................................................................................ 10

5.3 Analysis of Variance ...................................................................................................................... 10

5.4 Effect Tests ..................................................................................................................................... 11

5.5 Prediction Profiler ......................................................................................................................... 12

5.6 Residual Plots................................................................................................................................. 13

5.6.1 Residual by Predicted Plot.......................................................................................................... 13

5.6.2 Residual by Row plot .................................................................................................................. 13

5.7 Interaction Profiles ........................................................................................................................ 14

5.8 Normal Plot .................................................................................................................................... 15

5.9 Plot of Residuals versus Factors .................................................................................................... 16

6.0 Conclusion...................................................................................................................................... 18

7.0 References ...................................................................................................................................... 19



Figures Figure 1: Chord Length ............................................................................................................................ 4 Figure 2: Maximum Airfoil Thickness ...................................................................................................... 5 Figure 3: Dihedral Angle.......................................................................................................................... 5 Figure 4: Wing Sweep ............................................................................................................................ 6 Figure 5: Actual by Predicted Plot .......................................................................................................... 9 Figure 6 : Prediction Profiler ................................................................................................................. 12 Figure 7: Residual by Prediction Plot .................................................................................................... 13 Figure 8: Residual by Row Plot .............................................................................................................. 13 Figure 9: Interaction Profiles ................................................................................................................. 14 Figure 10: Normal Plot .......................................................................................................................... 17 Figure 11: Plot of Residual vs Chord length .......................................................................................... 16 Figure 12: Plot of Residual vs Maximum Airfoil Thickness .................................................................... 16 Figure 13: Plot of Residual vs Dihedral Angle ........................................................................................ 17 Figure 14: Plot of Residual vs Sweep Angle ........................................................................................... 17 Figure 15: Plot of Residual vs Blocks ..................................................................................................... 18

Tables Table 1: Factor Type and Levels .............................................................................................................. 7 Table 2: Design Matrix ............................................................................................................................ 5 Table 3: Summary of Fit .......................................................................................................................... 5 Table 4: Analysis of Variance .................................................................................................................. 9 Table 5 : Parameter Estimates .............................................................................................................. 12 Table 6: Effect Tests .............................................................................................................................. 13 Table 7: Sorted Parameter Estimates .................................................................................................... 14



1. INTRODUCTION

Free flight glider

Free flight gliders are scaled down model of aircrafts. They are unpowered aircrafts which means that they do not have any motors or propulsion systems to power the aircraft. Hence its flight is primarily dependent on its wings which produce the lift. As the glider moves faster through the air, the body and the wing of the glider produces drag which affects the flight duration. It is therefore essential to build free flight gliders that are aerodynamically efficient to obtain maximum flight duration.

Free flight gliders are usually made using light-weight materials such as balsa wood, styro-foam, fiber glass, etc. Free flight gliders are launched predominantly by hand. They are also launched using elastic bands.

Free flight glider modeling

Building free flight glider is a famous hobby with the primary challenge being building the most efficient glider that yields maximum flight duration. There are many competitions held around the world, notably the World Championships held by FAI (Fédération Aéronautique Internationale), a governing body for air sports and aeronautics world records.

1. PRE - EXPERIMENTAL PLANNING

2.1 Recognition of and statement of the problem

This project deals with the design and analysis of the experiment involving free flight gliders that have different wing geometry and dimensions. This project primarily focuses on identifying the factors that influence the flight duration of a free flight glider and possibly obtaining the optimal combination of wing configuration and wing dimensions that will result in maximization of flight duration. The sole objective of a free flight glider is to render maximum flight duration. In this experiment, the wing configuration and wing dimensions are varied to find the optimal combination. The geometry and dimensions of the body and tail section are kept constant.



2.2 Selection of Response Variable

The Response Variables that can be considered for this experiment are flight duration and the distance covered by the glider. Since the flight path is not linear always and unpredictable, the most suitable response variable is flight duration. Hence, flight duration (in seconds) is selected as the response variable for this experiment.

The glider is hand launched and the flight duration is measured from the instant the glider is launched to the instant the glider touches the ground. Time is measured using a digital stop clock.

2.3 Choice of factors, levels, and range

Wing is the most primary component of the glider that produces lift and helps the glider sustain its flight. In this experiment we are varying the wing configuration and wing dimensions to find out the factors that affect the flight duration and possibly obtain the optimal combination that produces maximum flight duration.

2.3.1 DESIGN FACTORS

1. Chord Length Chord length is the distance between the leading edge and trailing edge of airfoil. Levels in this factor:

High chord length (4.5 cm ) Low chord length ( 3 cm )

Figure 1: Chord length



2. Maximum Airfoil thickness Airfoil is the cross sectional shape of the wing. Levels in this factor:

High maximum airfoil thickness ( 8mm) Low maximum airfoil thickness (6mm)

Figure 2: Maximum Airfoil Thickness

3. Dihedral Angle It is the angle the wing makes with a horizontal reference. Levels in this factor:

No dihedral angle ( 0˚) Positive dihedral angle ( 5˚)

Figure 3: Dihedral Angle

4. Wing sweep The wing angles backward from the root of the wing to the tip of the wing. Levels in this factor:

Straight ( No wing sweep ) Swept back ( 20˚ )



Figure 4: Wing Sweep

2.3.2 HELD CONSTANT FACTORS

1. Wingspan ( 18 cm ) It is the distance between the two wing tips. The wingspan of the glider wing is kept constant for the entire experiment.

2. Body and Tail section The geometry and dimensions of the body and tail section are kept constant for the entire experiment.

2.3.3 UNCONTROLLABLE FACTORS

Wind gust Thermal current 3. CHOICE OF EXPERIMENTAL DESIGN

3.1 Design

It is feasible to perform the entire runs of this experiment; hence the experiment is run using a full factorial design. In this experiment, there are four design factors under consideration with two levels assigned to each design factor. Therefore it is a 24full factorial design. Since the glider is being launched by a person, it is ideal to run the experiment in two blocks with different operators assigned to each block. Introduction of blocks in this experiment is to eliminate any nuisance source of variability that adversely affects the statistical analysis among the levels of the factors.

Number of replicates : 2 Number of runs : 32 Center points : No center points Blocks : 2



3.2 Factor Type and Levels

Factors Factor Type High Level Low Level

Airfoil Thickness Categorical 8 mm 6 mm

Chord Length Categorical 4.5 cm 3 cm

Dihedral Angle Categorical 0˚ 5˚

Sweep Angle Categorical 0˚ 20˚ In JMP software the factor type was chosen to be categorical as we are interested only in the values at the high and low levels and not any continuous range of values between limits.

Table 1: Factor Type and Levels

4. EXPERIMENTAL PROCEDURE

4.1 Glider Construction

The Free Flight Glider is built using balsa wood. The geometry and dimensions of the body and tail section of the glider for all the sixteen models are kept to be constant. Sixteen different wings are built obtained by varying the design factors (The combination of factors is obtained from the design matrix). The Wings are attached the body of the glider whose dimensions are identical to obtain sixteen different models of the free flight glider. The airfoil shape of the wing is selected to be asymmetrical and it obtained by shaping the balsa wood used for wings with emery sheet. The body, tail and wings of the aircraft are attached together with the help of super glue. The sharp edges in the model are chamfered to reduce the drag.

4.2 Experimental Run

Runs are made as per the design matrix run order generated from JMP software (which is randomized to reduce the variations caused by the experimental pattern). In each run, the flight duration is noted which is the response variable. The flight duration is the time (in seconds) from the moment the glider is launched to the moment it touches the ground. It is noted using a digital stop clock. The experiment is run in two separate blocks with the glider being launched by different operators for each block. Each operator is allowed two trials for each factor combination and the average of the two values is taken as the observation.

4.3 Design Matrix

The design matrix was obtained using JMP 10 software. Randomization was performed in JMP software. The experiment is run in two blocks. The response of the experiment is displayed in the Flight Duration Column.



Table 2: Design Matrix



5. RESULTS AND STATISTICAL ANALYSIS

5.1 Actual response versus Predicted response

The figure below shows the Actual versus the Predicted response (Flight duration)

Figure 5: Actual by Predicted Plot

The figure above shows that the regression line and the 95% confidence curves cross the sample mean, hence it indicates that the model explains a substantial proportion of the response yield variation.



5.2 R-Square

The various R-Square values are enumerated below:

Table 3: Summary of Fit

We use the R2 term to measure the total variability proportion suggested by the model. This particular model has an R2 value of 0.883518 which indicates that there is approximately 88% of variation in the observations. The adjusted R2 value is 0.819452. These R2 values are relatively good. The adjusted R2 value is slightly lower than the R2 value probably because of the presence of a non-significant factor as indicated by the analysis. Root mean square value of 0.675023 is generally the variation in the response attributed to random errors.

5.3 Analysis of Variance

Table 4: Analysis of Variance

The ANOVA gives us an F ratio value of 13.7909 and a prob>F value of <.001 which indicates that the model is significant.



Table 5: Parameter Estimates

5.4 Effect Tests

Table 6: Effect Tests

The results from the Effect tests run using JMP 10 software is shown above. In the above table, the values of Sum of squares, F Ratio and p-values for the main effects and the two factor interactions are enumerated. From the results obtained it is indicated that the factors: Chord length, Maximum airfoil thickness and Dihedral angle are significant. The interaction between Chord length and dihedral angle is also found to be significant. The significance of these effects are evident from their p-values all of which are <= 0.05. All these are indicated by their respective p-values which are highlighted using ’*’ symbol. The block effect was not found to be significant as opposed to our expectation before commencing the experiment. This indicates that response was less affected by different operators performing the experiment.



Table 7: Sorted Parameter Estimates

5.5 Prediction Profiler

Figure 6: Prediction Profiler

From the prediction profiler shown above, it is evident that keeping the chord length at high level (4.5) significantly increases the flight duration with greater magnitude as compared to other factors. It also indicates that keeping the maximum airfoil thickness at high level (0.8) slightly reduces the response (flight duration). Having a positive dihedral angle (5˚) increases the flight duration. The change in sweep angle has little negative or almost no effect on the response.



5.6 Residual Plots

5.6.1 Residual by Predicted Plot

Figure 7: Residual by Predicted Plot

The above figure displays the Residual by Predicted Plot. The above plot is structure-less and does not expose any noticeable pattern. Hence, the model is correct and the assumptions are satisfied. There is no visible evidence of non-constant variance as there is no increase in the variance of the observations as the scale of the response increases.

5.6.2 Residual by Row plot

Figure 8: Residual by Row Plot

The above plot of Residuals by Row plot does not disclose any issues as the variance does not indicate any specific pattern or show increase over the run order. There is no violation of constant variance or independence assumptions.



5.7 Interaction Profiles

The interaction profiles for the various factor levels are shown below:

Figure 9: Interaction Profiles

We see interaction between the levels of the sweep angle for the limits considered in the experiment. Few other factors like dihedral angle, maximum airfoil thickness and chord length do not show interaction in the limits considered in this experiment but will show significant interaction if the limits are extrapolated



5.8 Normal Plot

Figure 10: Normal Plot

The normal plot above signifies that the effects chord length (high level – 4.5cm), Dihedral angle (high level - 5˚), Maximum airfoil thickness (high level – 0.8 cm) and the interactions between chord length (high level – 4.5cm) & Dihedral angle (high level - 5˚) and chord length (high level – 4.5cm) & sweep angle (high level - 20˚) are the significant effects. But the statistical analysis indicates near significance of the chord length (high level – 4.5cm) & sweep angle (high level - 20˚) interaction.



5.9 Plot of Residuals versus Factors

Figure 11: Plot of Residual vs. Chord length

Figure 12: Plot of Residual vs. Maximum Airfoil Thickness



Figure 13: Plot of Residual vs. Dihedral Angle

Figure 14: Plot of Residual vs. Sweep Angle



Figure 15: Plot of Residual vs. Block

The above figures display the Plot of Residuals versus Individual factors. The plot of factors chord length, maximum airfoil thickness, dihedral angle, sweep angle and blocks versus the residuals have almost the same variability at both the levels of the factor. There is no evidence of any abnormality in the above graphs.

6. CONCLUSION

The objective of this experiment is to screen for the factors that affect the flight duration of the free flight glider and possibly arrive at a combination of factor levels that yield maximum response. From the statistical analysis of the experiment, it is inferred that it is desirable to have the factors chord length and dihedral angle at their high levels respectively. In the case of maximum airfoil thickness, it is advisable to keep the factor level low because keeping this factor at the high level adversely affects the flight duration. For the final main effect sweep angle, it is inferred from the analysis that this main effect is not significant and has little or effect on the response. With respect to interactions, we find that there is significant interaction between chord length and dihedral angle at their respective high factor levels.

Ideally the optimal combination of factors renders a wing having a chord length of 4.5 cm, maximum airfoil thickness of .6 cm and a configuration that has positive dihedral angle (5˚) and no wing sweep. Wing sweep though not indicated as significant in the analysis is preferred to be kept at the low factor level because the statistical analysis shows that high level interaction between wing sweep and chord length has negative effect that is close to significance. The above combination yields maximum glider flight duration for the given dimensions of body and tail section and the factor limits considered.



7. REFERENCES

Montgomery, C. Douglas, “Design and Analysis of Experiments”, John Wiley & Sons, Inc., 8th edition

http://en.wikipedia.org/wiki/Free_flight_(model_aircraft)

SAS Institute Inc. 2009. JMP® 8 Introductory Guide, Second Edition. Cary, NC: SAS Institute Inc. SAS Institute Inc. 2009. JMP® 8 Design of Experiments Guide, Second Edition. Cary, NC: SAS Institute Inc. Images: www.wikipedia.org

maximization of flight duration of a free flight glider (doe)

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