Statistical Calibration and Validation of Mathematical Model to Predict Motion of Paper Helicopter

종이 헬리콥터 낙하해석모델의 통계적 교정 및 검증

  • Kim, Gil Young (School of Mechanical and Aerospace Engineering, Korea Aerospace Univ.) ;
  • Yoo, Sung Bum (School of Mechanical and Aerospace Engineering, Korea Aerospace Univ.) ;
  • Kim, Dong Young (School of Mechanical and Aerospace Engineering, Korea Aerospace Univ.) ;
  • Kim, Dong Seong (School of Mechanical and Aerospace Engineering, Korea Aerospace Univ.) ;
  • Choi, Joo Ho (School of Mechanical and Aerospace Engineering, Korea Aerospace Univ.)
  • 김길영 (한국항공대학교 항공우주 및 기계공학부) ;
  • 유성범 (한국항공대학교 항공우주 및 기계공학부) ;
  • 김동영 (한국항공대학교 항공우주 및 기계공학부) ;
  • 김동성 (한국항공대학교 항공우주 및 기계공학부) ;
  • 최주호 (한국항공대학교 항공우주 및 기계공학부)
  • Received : 2014.10.10
  • Accepted : 2015.06.16
  • Published : 2015.08.01


Mathematical models are actively used to reduce the experimental expenses required to understand physical phenomena. However, they are different from real phenomena because of assumptions or uncertain parameters. In this study, we present a calibration and validation method using a paper helicopter and statistical methods to quantify the uncertainty. The data from the experiment using three nominally identical paper helicopters consist of different groups, and are used to calibrate the drag coefficient, which is an unknown input parameter in both analytical models. We predict the predicted fall time data using probability distributions. We validate the analysis models by comparing the predicted distribution and the experimental data distribution. Moreover, we quantify the uncertainty using the Markov Chain Monte Carlo method. In addition, we compare the manufacturing error and experimental error obtained from the fall-time data using Analysis of Variance. As a result, all of the paper helicopters are treated as one identical model.


Paper Helicopter;Predictive Validation;Markov Chain Monte-Carlo Simulation;Statistical Calibration;Analysis of Variance


  1. Bayarri, M.J., Berger, J.O., Higdon, D., Kennedy, M.C., Kottas, A., Paulo, R., Sacks, J., Cafeo, J.A., Cavendish, J., Lin, C.H. and Tu, J., 2002, "A Framework for Validation of Computer Models," National Institute of Statistical Sciences.
  2. Siorek, T. and Haftka, R.T., 1998, "Paper Helicopter -Experimental Optimum Engineering Design Classroom Problem," AIAA Paper 98-4963 Proceedings, 7th AIAA/USAF/NASA/-ISSMO Symposium on Multidisciplinary Analysis and Optimization, pp. 2013-2020, St.Louis MO, Sept. 2-4.
  3. Annis, D., 2005, "Rethinking the Paper Helicopter: Combining Statistical and Engineering Knowledge," The American Statistician, 59, No. 4, pp. 320-326.
  4. LaTourette, K.J., "The Application of Differential Equations to Model the Motion of a Paper Helicopter," Saint John Fisher College
  5. Park, C., Choi, J.-h. and Haftka, R.T., 2014, "Teaching a Verification and Validation Course Using Simulations and Experiments with Paper Helicopter," ASME 2014 Verification and Validation Symposium
  6. Oberkampf and Barone, 2004
  7. Yoo, M.Y. and Choi, J.H., 2013, "Probabilistic Calibration of Computer Model and Application to Reliability Analysis of Elasto-Plastic Insertion Problem," Trans. Korean Soc. Mech. Eng. A, Vol. 37, No. 9, pp. 1133-1140.
  8. Ferson, S., Oberkampf, W. L. and Ginzburg, L., 2008, Model Validation and Predictive Capability for the Thermal Challenge Problem. Computer Methods in Applied Mechanics and Engineering, 197(29), pp. 2408-2430.
  9. Minitab. Retrieved from