Mathematical Modeling

  • Ranjan GanguliEmail author
  • Dipali Thakkar
  • Sathyamangalam Ramanarayanan Viswamurthy
Part of the Advances in Industrial Control book series (AIC)


The governing equations of the helicopter rotor blade enabling piezoceramic-axial, bending, and shear actuation are derived in this chapter and an outline of the aeroelastic analysis used in this book is given. A background on piezoelectric materials used in this book is also provided. Section 2.1 begins with an introduction to piezoelectric materials and Sect. 2.2 explains the piezoceramic actuation concept. Section 2.3 provides an introduction to terminology used in the helicopter field. In Sect. 2.4, structural modeling is explained. Section 2.5 explains the aerodynamic model used for the aeroelastic analysis. Section 2.6 presents the blade and hub loads. Section 2.7 explains the aeroelastic analysis of a rotor. Section 2.8 gives the summary of this chapter.


Piezoelectric Material Rotor Blade Rotor Disk Forward Speed Aerodynamic Model 
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  1. 1.
    Durr, J., Schmidt, U., Zaglauer, W.: On the integration of piezoceramic actuators in composite structures for aerospace applications. J. Intell. Mater. Syst. Struct. 10, 880–889 (1999)CrossRefGoogle Scholar
  2. 2.
    Srinivasan, A., McFarland, M.: Smart Structures: Analysis and Design. Cambridge University Press, Cambridge (2000)Google Scholar
  3. 3.
    Crawley, E.: Intelligent structures for aerospace: a technology overview and assessment. J. Am. Inst. Aeronaut. Astronaut. 32(8), 1689–1699 (1994)Google Scholar
  4. 4.
    Loewy, R.: Recent developments in smart structures with aeronautical applications. Smart Mater. Struct. 6(5), 11–42 (1997)CrossRefGoogle Scholar
  5. 5.
    Park, S., Shrout, T.: Relaxor based ferroelectric single crystals for electromechanical actuators. Mater. Res. Innov. 1, 20–25 (1997)CrossRefGoogle Scholar
  6. 6.
    Liu, S., Ren, W., Mukherjee, B.: The piezoelectric shear strain coefficient of \({<}111{>}\) piezocrystals. Appl. Phys. Lett. 83(14), 2886–2888 (2003)Google Scholar
  7. 7.
    Crawley, E., Anderson, E.: Detailed models of piezoceramic actuation of beams. In: Proceedings of the 30th AIAA/ASMI/ASCH/AHS/ASC Structures, Structural Dynamics Conference, Washington, DC (1989)Google Scholar
  8. 8.
    Hong, C., Chopra, I.: Modeling and validation of induced strain actuation of composite coupled plates. J. Am. Inst. Aeronaut. Astronaut. 37(3), 372–377 (1999)CrossRefGoogle Scholar
  9. 9.
    Chen, W., Saleeb, A.: Constitutive Equations for Engineering Materials: Elasticity and Modeling, vol. 1. Wiley-Interscience, New York (1952)zbMATHGoogle Scholar
  10. 10.
    Hodges, D., Dowell, E.: Non-linear equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades. Technical report NASA TN D-7818 (1974)Google Scholar
  11. 11.
    Epps, J., Chandra, R.: The natural frequencies of rotating composite beams with tip sweep. J. Am. Helicopter Soc. 41, 29–36 (1996)Google Scholar
  12. 12.
    Ganguli, R., Chopra, I., Weller, W.: Comparison of calculated vibratory rotor hub loads with experimental data. J. Am. Helicopter Soc. 43, 312–318 (1998)CrossRefGoogle Scholar
  13. 13.
    Chopra, I., Sivaneri, T.: Aeroelastic stability of rotor blades using finite element analysis. Technical report NASA CR 166389 (1982)Google Scholar
  14. 14.
    Leishman, J.G., Beddoes, T.S.: A generalised model for airfoil unsteady aerodynamic behaviour and dynamic stall using the indicial method. In: Proceedings of the 42nd Annual Forum of the American Helicopter Society, Washington, DC, June, pp. 243–265 (1986)Google Scholar
  15. 15.
    Leishman, J.G., Beddoes, T.S.: A semi-empirical model for dynamic stall. J. Am. Helicopter Soc. 34(3), 3–17 (1989); Leishman, J.G.: Principles of Helicopter Aerodynamics. Cambridge University Press, New York (2000)Google Scholar
  16. 16.
    Drees, J.M.: A theory of air flow through rotors and its application to some helicopter problems. J. Helicopter Assoc. G. B. 3(2), 79–104 (1949)Google Scholar
  17. 17.
    Bagai, A., Leishman, J.G.: Rotor free-wake modeling using a pseudo-implicit technique—including comparisons with experimental data. J. Am. Helicopter Soc. 40(3), 29–41 (1995)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Ranjan Ganguli
    • 1
    Email author
  • Dipali Thakkar
    • 2
  • Sathyamangalam Ramanarayanan Viswamurthy
    • 3
  1. 1.Department of Aerospace EngineeringIndian Institute of ScienceBangaloreIndia
  2. 2.Department of Aeronautical EngineeringSardar Vallabhbhai Patel Institute of TechnologyVasadIndia
  3. 3.Advanced Composites DivisionCSIR-National Aerospace LaboratoriesBangaloreIndia

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