Research Papers

Sensitivity of Lyapunov Exponents in Design Optimization of Nonlinear Dampers

[+] Author and Article Information
Aykut Tamer

Department of Aerospace
Science and Technology,
Politecnico Di Milano,
Milano 20156, Italy
e-mail: aykut.tamer@polimi.it

Pierangelo Masarati

Department of Aerospace
Science and Technology,
Politecnico Di Milano,
Milano 20156, Italy
e-mail: pierangelo.masarati@polimi.it

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received June 7, 2018; final manuscript received October 21, 2018; published online January 7, 2019. Assoc. Editor: Radu Serban.

J. Comput. Nonlinear Dynam 14(2), 021002 (Jan 07, 2019) (9 pages) Paper No: CND-18-1263; doi: 10.1115/1.4041827 History: Received June 07, 2018; Revised October 21, 2018

This work presents how the analytical sensitivity of Lyapunov characteristic exponents (LCEs) can be used in the design of nonlinear dampers, which are frequently utilized to stabilize the response of mechanical systems. The kinetic energy dissipated in the form of heat often induces nonlinearities, therefore reducing the reliability of standard stability evaluation methods. Owing to the difficulty of estimating the stability properties of equilibrium solution of the resulting nonlinear time-dependent systems, engineers usually tend to linearize and time-average the governing equations. However, the solutions of nonlinear and time-dependent dynamical systems may exhibit unique properties, which are lost when they are simplified. When a damper is designed based on a simplified model, the cost associated with neglecting nonlinearities can be significantly high in terms of safety margins that are needed as a safeguard with respect to model uncertainties. Therefore, in those cases, a generalized stability measure, with its parametric sensitivity, can replace usual model simplifications in engineering design, especially when a system is dominated by specific, non-negligible nonlinearities and time-dependencies. The estimation of the characteristic exponents and their sensitivity is illustrated. A practical application of the proposed methodology is presented, considering that the problem of helicopter ground resonance (GR) and landing gear shimmy vibration with nonlinear dampers are implemented instead of linear ones. Exploiting the analytical sensitivity of the Lyapunov exponents within a continuation approach, the geometric parameters of the damper are determined. The mass of the damper and the largest characteristic exponent of the system are used as the objective function and the inequality or equality constraint in the design of the viscous dampers.

Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.


Hirsch, M. W. , Smale, S. , and Devaney, R. L. , 2004, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Elsevier, San Diego, CA.
Medio, A. , and Lines, M. , 2001, Nonlinear Dynamics—A Primer, Cambridge University Press, Cambridge, UK.
Strogatz, S. H. , 1994, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Perseus Books, Reading, MA.
Adrianova, L. Y. , 1995, Introduction to Linear Systems of Differential Equations (Translations of Mathematical Monographs, Vol. 146), American Mathematical Society, Providence, RI.
Hodges, D. , and Pierce, G. A. , 2002, Introduction to Structural Dynamics and Aeroelasticity, Cambridge University Press, Cambridge, UK.
Dowell, E. H. , 2015, A Modern Course in Aeroelasticity, 5th ed., Springer, Cham, Switzerland.
Benettin, G. , Galgani, L. , Giorgilli, A. , and Strelcyn, J.-M. , 1980, “ Lyapunov Characteristic Exponents for Smooth Dynamical Systems and for Hamiltonian Systems; A Method for Computing All of Them—Part 1: Theory,” Meccanica, 15(1), pp. 9–20. [CrossRef]
Cong, N. D. , and Nam, H. , 2003, “ Lyapunov's Inequality for Linear Differential Algebraic Equation,” Acta Math. Vietnam., 28(1), pp. 73–88. http://journals.math.ac.vn/acta/pdf/0301073.pdf
Cong, N. D. , and Nam, H. , 2004, “ Lyapunov Regularity of Linear Differential Algebraic Equations of Index 1,” Acta Math. Vietnam., 29(1), pp. 1–21. http://journals.math.ac.vn/acta/pdf/0401001.pdf
Masarati, P. , 2013, “ Estimation of Lyapunov Exponents From Multibody Dynamics in Differential-Algebraic Form,” Proc. IMechE Part K: J. Multi-Body Dyn., 227(1), pp. 23–33.
Rao, S. , 1996, Engineering Optimization: Theory and Practice, 3rd ed., Wiley, Hoboken, NJ.
Krauskopf, B. , Osinga, H. M. , and Galán-Vioque, J. , 2007, Numerical Continuation Methods for Dynamical Systems, Springer, Dordrecht, The Netherlands.
Shih, I.-C. , Spence, A. M. , and Celi, R. , 1996, “ Semianalytical Sensitivity of Floquet Characteristic Exponents With Application to Rotary-Wing Aeroelasticity,” J. Aircr., 33(2), pp. 322–330. [CrossRef]
Masarati, P. , and Tamer, A. , 2015, “ Sensitivity of Trajectory Stability Estimated by Lyapunov Characteristic Exponents,” Aerosp. Sci. Technol., 47, pp. 501–510. [CrossRef]
Tamer, A. , and Masarati, P. , 2016, “ Stability of Nonlinear, Time-Dependent Rotorcraft Systems Using Lyapunov Characteristic Exponents,” J. Am. Helicopter Soc., 61(2), pp. 1–12. [CrossRef]
Dieci, L. , and Van Vleck, E. S. , 2002, “ Lyapunov Spectral Intervals: Theory and Computation,” SIAM J. Numer. Anal., 40(2), pp. 516–542. [CrossRef]
Coleman, R. P. , and Feingold, A. M. , 1958, “ Theory of Self-Excited Mechanical Oscillations of Helicopter Rotors With Hinged Blades,” National Advisory Committee for Aeronautics, Langley Field, VA, Report No. NACA-TR-1351.
Hammond, C. E. , 1974, “ An Application of Floquet Theory to Prediction of Mechanical Instability,” J. Am. Helicopter Soc., 19(4), pp. 14–23. [CrossRef]
Bir, G. , 2008, “ Multiblade Coordinate Transformation and Its Application to Wind Turbine Analysis,” AIAA Paper No. 2008-1300.
Tourajizadeh, H. , and Zare, S. , 2016, “ Robust and Optimal Control of Shimmy Vibration in Aircraft Nose Landing Gear,” Aerosp. Sci. Technol., 50, pp. 1–14. [CrossRef]
Li, Y. , Jiang, J. Z. , and Neild, S. , 2016, “ Optimisation of Shimmy Suppression Device in an Aircraft Main Landing Gear,” J. Phys.: Conf. Ser., 744(1), p. 012066. [CrossRef]
Somieski, G. , 1997, “ Shimmy Analysis of a Simple Aircraft Nose Landing Gear Model Using Different Mathematical Methods,” Aerosp. Sci. Technol., 1(8), pp. 547–555. [CrossRef]
Besselink, I. , 2000, “ Shimmy of Aircraft Main Landing Gears,” Ph.D. thesis, TU Delft, Delft, The Netherlands.
Arreaza, C. , 2015, “ Linear Stability Analysis, Dynamic Response, and Design of Shimmy Dampers for Main Landing Gears,” Master's thesis, University of Toronto, Toronto, ON, Canada.
Jugulkar, L. M. , Singh, S. , and Sawant, S. M. , 2016, “ Analysis of Suspension With Variable Stiffness and Variable Damping Force for Automotive Applications,” Adv. Mech. Eng., 8(5), pp. 1–19.
Marathe, S. , Gandhi, F. , and Wang, K. W. , 1998, “ Helicopter Blade Response and Aeromechanical Stability With a Magnetorheological Fluid Based Lag Damper,” J. Intell. Mater. Syst. Struct., 9(4), pp. 272–282. [CrossRef]
Li, Z. Q. , Xu, Y. L. , and Zhou, L. M. , 2006, “ Adjustable Fluid Damper With Sma Actuators,” Smart Mater. Struct., 15(5), p. 1483. [CrossRef]
Tamer, A. , and Masarati, P. , 2013, “ Linearized Structural Dynamics Model for the Sensitivity Analysis of Helicopter Rotor Blades,” Ankara International Aerospace Conference, Ankara, Turkey, Sept. 11–13, pp. 1–11.
Tamer, A. , and Masarati, P. , 2014, “ Periodic Stability and Sensitivity Analysis of Rotating Machinery,” Ninth IFToMM International Conference on Rotor Dynamics (ICORD 2014), Milan. Switzerland, Sept. 24–25, pp. 2059–2070.
Cardani, C. , and Mantegazza, P. , 1976, “ Continuation and Direct Solution of the Flutter Equation,” Comput. Struct., 8(2), pp. 185–192.
Kim, I. Y. , and Kwak, B. M. , 2002, “ Design Space Optimization Using a Numerical Design Continuation Method,” Int. J. Numer. Methods Eng., 53(8), pp. 1979–2002. [CrossRef]
Rezgui, D. , Lowenberg, M. H. , Jones, M. , and Monteggia, C. , 2014, “ Continuation and Bifurcation Analysis in Helicopter Aeroelastic Stability Problems,” J. Guid., Control, Dyn., 37(3), pp. 889–897. [CrossRef]
Jones, M. , Bernascone, A. , Masarati, P. , Quaranta, G. , and Rezgui, D. , 2014, “ Ongoing Developments in the Use of Continuation-Bifurcation Methodology at AgustaWestland,” 40th European Rotorcraft Forum, Southampton, UK, Sept. 2–5, pp. 1–14.


Grahic Jump Location
Fig. 1

Sketch of Hammond's helicopter GR model with one blade is presented for clarity (a) and typical collocation of lag damper (b): (a) Hammond's model (b) offset from lag hinge

Grahic Jump Location
Fig. 2

Idealized yaw motion dynamics of a landing gear

Grahic Jump Location
Fig. 3

Cross section of a typical viscous damper

Grahic Jump Location
Fig. 4

The analytical sensitivity of LCEs to linear damping coefficient is compared and validated using discrete derivative of previously calculated LCEs: (a) 2 largest LCEs and (b) sensitivity of 2 largest LCEs

Grahic Jump Location
Fig. 5

Ground resonance: design of blade damper for a minimum mass while ensuring the value of largest LCE estimate at a given value. Initial and desired values are m0 = 10 kg, mmin = 3 kg, and λmax = −0.5 rad s−1.

Grahic Jump Location
Fig. 6

Shimmy vibration: design of a shimmy damper for minimum mass while ensuring the value of the largest LCE estimate at a given value. Initial and desired values are m0 = 6 kg, mmin = 3 kg, and λmax = −1.0 rad s−1.



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In