Skip to main content
Book cover

Dynamic Stability and Bifurcation in Nonconservative Mechanics pp 129–190Cite as

Classical Results and Modern Approaches to Nonconservative Stability

Part of the CISM International Centre for Mechanical Sciences book series (CISM,volume 586)

Abstract

Stability of nonconservative systems is nontrivial already on the linear level, especially, if the system depends on multiple parameters. We present an overview of results and methods of stability theory that are specific for nonconservative applications. Special attention is given to the topics of flutter and divergence, reversible- and Hamiltonian-Hopf bifurcation, Krein signature, modes and waves of positive and negative energy, dissipation-induced instabilities, destabilization paradox, influence of structure of forces on stability and stability optimization.

This is a preview of subscription content, access via your institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-93722-9_4
  • Chapter length: 62 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   129.00
Price excludes VAT (USA)
  • ISBN: 978-3-319-93722-9
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   169.99
Price excludes VAT (USA)
Hardcover Book
USD   169.99
Price excludes VAT (USA)
Learn about institutional subscriptions
Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26

References

  • N. Andersson, Gravitational waves from instabilities in relativistic stars. Class. Quantum Grav. 20, R105–R144 (2003)

    MathSciNet  CrossRef  Google Scholar 

  • I.P. Andreichikov, V.I. Yudovich, The stability of visco-elastic rods. Izv. Akad. Nauk SSSR. Mekhanika Tverdogo Tela. 9(2), 78–87 (1974)

    Google Scholar 

  • S. Aoi, Y. Egi, K. Tsuchiya, Instability-based mechanism for body undulations in centipede locomotion. Phys. Rev. E 87, 012717 (2013)

    CrossRef  Google Scholar 

  • V.I. Arnold, Lectures on bifurcations in versal families. Russ. Math. Surv. 27, 54–123 (1972)

    CrossRef  Google Scholar 

  • G.L. Austin Sydes, Self-stable bicycles. Bsc (Hons) mathematics final year project report. (Northumbria University, Newcastle upon Tyne, UK, 2018)

    Google Scholar 

  • P.V. Bayly, S.K. Dutcher, Steady dynein forces induce flutter instability and propagating waves in mathematical models of flagella. J. R. Soc. Interface 13, 20160523 (2016)

    CrossRef  Google Scholar 

  • M. Beck, Die Knicklast des einseitig eingespannten, tangential gedruckten Stabes. Z. angew. Math. Phys. 3, 225–228 (1952)

    CrossRef  Google Scholar 

  • V.V. Beletsky, Some stability problems in applied mechanics. Appl. Math. Comput. 70, 117–141 (1995)

    MathSciNet  MATH  Google Scholar 

  • M.V. Berry, P. Shukla, Curl force dynamics: symmetries, chaos and constants of motion. New J. Phys. 18, 063018 (2016)

    CrossRef  Google Scholar 

  • D. Bigoni, G. Noselli, Experimental evidence of flutter and divergence instabilities induced by dry friction. J. Mech. Phys. Sol. 59, 2208–2226 (2011)

    CrossRef  Google Scholar 

  • D. Bigoni, D. Misseroni, M. Tommasini, O.N. Kirillov, G. Noselli, Detecting singular weak-dissipation limit for flutter onset in reversible systems. Phys. Rev. E 97(2), 023003 (2018)

    CrossRef  Google Scholar 

  • A.M. Bloch, P.S. Krishnaprasad, J.E. Marsden, T.S. Ratiu, Dissipation induced instabilities. Annales de L’Institut Henri Poincare - Analyse Non Lineaire 11, 37–90 (1994)

    MathSciNet  CrossRef  Google Scholar 

  • V.V. Bolotin, Nonconservative Problems of the Theory of Elastic Stability (Pergamon Press, Oxford, 1963)

    MATH  Google Scholar 

  • A.V. Borisov, A.A. Kilin, I.S. Mamaev, The Hamiltonian dynamics of self-gravitating liquid and gas ellipsoids. Reg. Chaotic Dyn. 14(2), 179–217 (2009)

    MathSciNet  CrossRef  Google Scholar 

  • O. Bottema, On the stability of the equilibrium of a linear mechanical system, ZAMP Z. Angew. Math. Phys. 6, 97–104 (1955)

    MathSciNet  CrossRef  Google Scholar 

  • O. Bottema, The Routh-Hurwitz condition for the biquadratic equation. Indag. Math. (Proc.) 59, 403–406 (1956)

    MathSciNet  CrossRef  Google Scholar 

  • R.M. Bulatovic, A sufficient condition for instability of equilibrium of nonconservative undamped systems. Phys. Lett. A 375, 3826–3828 (2011)

    MathSciNet  CrossRef  Google Scholar 

  • R.M. Bulatovic, A stability criterion for circulatory systems. Acta Mech. 228(7), 2713–2718 (2017)

    MathSciNet  CrossRef  Google Scholar 

  • S. Chandrasekhar, Ellipsoidal Figures of Equilibrium (Yale University Press, New Haven, 1969)

    MATH  Google Scholar 

  • S. Chandrasekhar, Solutions of two problems in the theory of gravitational radiation. Phys. Rev. Lett. 24(11), 611–615 (1970)

    CrossRef  Google Scholar 

  • S. Chandrasekhar, On stars, their evolution and their stability. Science 226(4674), 497–505 (1984)

    CrossRef  Google Scholar 

  • G. De Canio, E. Lauga, R.E. Goldstein, Spontaneous oscillations of elastic filaments induced by molecular motors. J. R. Soc. Interface 14, 20170491 (2017)

    CrossRef  Google Scholar 

  • P. Gallina, About the stability of non-conservative undamped systems. J. Sound Vibr. 262, 977–988 (2003)

    MathSciNet  CrossRef  Google Scholar 

  • V.L. Ginzburg, V.N. Tsytovich, Several problems of the theory of transition radiation and transition scattering. Phys. Rep. 49(1), 1–89 (1979)

    CrossRef  Google Scholar 

  • G. Gladwell, Follower forces - Leipholz early researches in elastic stability. Can. J. Civil Eng. 17, 277–286 (1990)

    CrossRef  Google Scholar 

  • A.G. Greenhill, On the rotation required for the stability of an elongated projectile. Min. Proc. R. Artill. Inst. X(7), 577–593 (1879)

    Google Scholar 

  • A.G. Greenhill, On the general motion of a liquid ellipsoid under the gravitation of its own parts. Proc. Camb. Philos. Soc. 4, 4–14 (1880)

    MATH  Google Scholar 

  • A.G. Greenhill, Determination of the greatest height consistent with stability that a vertical pole or must can be made, and of the greatest height to which a tree of given proportions can grow. Proc. Camb. Philos. Soc. 4, 65–73 (1881)

    MATH  Google Scholar 

  • A.G. Greenhill, On the strength of shafting when exposed both to torsion and to end thrust. Proc. Inst. Mech. Eng. 34, 182–225 (1883)

    CrossRef  Google Scholar 

  • P. Hagedorn, E. Heffel, P. Lancaster, P.C. Müller, S. Kapuria, Some recent results on MDGKN-systems. ZAMM - Z. Angew. Math. Mech. 95(7), 695–702 (2014)

    MathSciNet  CrossRef  Google Scholar 

  • P.L. Kapitsa, Stability and passage through the critical speed of the fast spinning rotors in the presence of damping. Z. Tech. Phys. 9, 124–147 (1939)

    Google Scholar 

  • M.A. Karami, D.J. Inman, Equivalent damping and frequency change for linear and nonlinear hybrid vibrational energy harvesting systems. J. Sound Vibr. 330, 5583–5597 (2011)

    CrossRef  Google Scholar 

  • A.L. Kimball, Internal friction as a cause of shaft whirling. Phil. Mag. 49, 724–727 (1925)

    CrossRef  Google Scholar 

  • O.N. Kirillov, Gyroscopic stabilization in the presence of nonconservative forces. Doklady Math. 76(2), 780–785 (2007)

    MathSciNet  CrossRef  Google Scholar 

  • O.N. Kirillov, Campbell diagrams of weakly anisotropic flexible rotors. Proc. R. Soc. A 465(2109), 2703–2723 (2009)

    MathSciNet  CrossRef  Google Scholar 

  • O.N. Kirillov, Eigenvalue bifurcation in multiparameter families of non-self-adjoint operator matrices. ZAMP - Z. Angew. Math. Phys. 61, 221–234 (2010)

    MathSciNet  CrossRef  Google Scholar 

  • O.N. Kirillov, Sensitivity of sub-critical mode-coupling instabilities in non-conservative rotating continua to stiffness and damping modifications. Int. J. Vehicle Struct. Syst. 3(1), 1–13 (2011a)

    Google Scholar 

  • O.N. Kirillov, Brouwer’s problem on a heavy particle in a rotating vessel: wave propagation, ion traps, and rotor dynamics. Phys. Lett. A 375, 1653–1660 (2011b)

    CrossRef  Google Scholar 

  • O.N. Kirillov, Nonconservative Stability Problems of Modern Physics (De Gruyter, Berlin, 2013a)

    Google Scholar 

  • O.N. Kirillov, Stabilizing and destabilizing perturbations of PT-symmetric indefinitely damped systems. Phil. Trans. R. Soc. A 371, 20120051 (2013b)

    MathSciNet  CrossRef  Google Scholar 

  • O.N. Kirillov, Singular diffusionless limits of double-diffusive instabilities in magnetohydrodynamics. Proc. R. Soc. A 473(2205), 20170344 (2017)

    MathSciNet  CrossRef  Google Scholar 

  • O.N. Kirillov, A.P. Seyranian, Metamorphoses of characteristic curves in circulatory systems. J. Appl. Math. Mech. 66, 371–385 (2002a)

    MathSciNet  CrossRef  Google Scholar 

  • O.N. Kirillov, A.P. Seyranian, A nonsmooth optimization problem. Moscow Univ. Mech. Bull. 57, 1–6 (2002b)

    MATH  Google Scholar 

  • O.N. Kirillov, F. Verhulst, Paradoxes of dissipation-induced destabilization or who opened Whitney’s umbrella? ZAMM - Z. Angew. Math. Mech. 90(6), 462–488 (2010)

    MathSciNet  CrossRef  Google Scholar 

  • W. Kliem, C. Pommer, A note on circulatory systems: old and new results. Z. Angew. Math. Mech. 97, 92–97 (2017)

    MathSciNet  CrossRef  Google Scholar 

  • J.D.G. Kooijman, J.P. Meijaard, J.M. Papadopoulos, A. Ruina, A.L. Schwab, A bicycle can be self-stable without gyroscopic or caster effects. Science 332(6027), 339–342 (2011)

    MathSciNet  CrossRef  Google Scholar 

  • N.D. Kopachevskii, S.G. Krein, Operator Approach in Linear Problems of Hydrodynamics. Self-adjoint Problems for an Ideal Fluid, Operator Theory: Advances and Applications, vol. 1 (Birkhauser, Basel, 2001)

    Google Scholar 

  • R. Krechetnikov, J.E. Marsden, Dissipation-induced instabilities in finite dimensions. Rev. Mod. Phys. 79, 519–553 (2007)

    MathSciNet  CrossRef  Google Scholar 

  • V. Lakhadanov, On stabilization of potential systems, Prikl. Mat. Mekh. 39, 53–58 (1975)

    MathSciNet  CrossRef  Google Scholar 

  • J.S.W. Lamb, J.A.G. Roberts, Time-reversal symmetry in dynamical systems: a survey. Phys. D 112, 1–39 (1998)

    MathSciNet  CrossRef  Google Scholar 

  • W.F. Langford, Hopf meets Hamilton under Whitney’s umbrella, in IUTAM Symposium on Nonlinear Stochastic Dynamics. Proceedings of the IUTAM Symposium, Monticello, IL, USA, Augsut 26–30, 2002, Solid Mech. Appl., vol. 110, ed. S.N. Namachchivaya, pp. 157–165 (Kluwer, Dordrecht, 2003)

    Google Scholar 

  • N.R. Lebovitz, Binary fission via inviscid trajectories. Geoph. Astroph. Fluid. Dyn. 38(1), 15–24 (1987)

    CrossRef  Google Scholar 

  • N.R. Lebovitz, The mathematical development of the classical ellipsoids. Int. J. Eng. Sci. 36(12), 1407–1420 (1998)

    MathSciNet  CrossRef  Google Scholar 

  • H. Leipholz, Stability Theory: an Introduction to the Stability of Dynamic Systems and Rigid Bodies, 2nd edn. (Teubner, Stuttgart, 1987)

    CrossRef  Google Scholar 

  • L. Lindblom, S.L. Detweiler, On the secular instabilities of the Maclaurin spheroids. Astrophys. J. 211, 565–567 (1977)

    CrossRef  Google Scholar 

  • A. Luongo, M. Ferretti, Postcritical behavior of a discrete Nicolai column. Nonlin. Dyn. 86, 2231–2243 (2016)

    MathSciNet  CrossRef  Google Scholar 

  • A. Luongo, M. Ferretti, F. D’Annibale, Paradoxes in dynamic stability of mechanical systems: investigating the causes and detecting the nonlinear behaviors. Springer Plus 5, 60 (2016)

    CrossRef  Google Scholar 

  • A.M. Lyapunov, The general problem of the stability of motion (translated into English by A. T. Fuller). Int. J. Control 55, 531–773 (1992)

    CrossRef  Google Scholar 

  • R.S. MacKay, Movement of eigenvalues of Hamiltonian equilibria under non-Hamiltonian perturbation. Phys. Lett. A 155, 266–268 (1991)

    MathSciNet  CrossRef  Google Scholar 

  • O. Mahrenholtz, R. Bogacz, On the shape of characteristic curves for optimal structures under non-conservative loads. Arch. Appl. Mech. 50, 141–148 (1981)

    MATH  Google Scholar 

  • S. Mandre, L. Mahadevan, A generalized theory of viscous and inviscid flutter. Proc. R. Soc. Lond. A 466, 141–156 (2010)

    MathSciNet  CrossRef  Google Scholar 

  • D.R. Merkin, Gyroscopic Systems (Gostekhizdat, Moscow, 1956) [in Russian]

    Google Scholar 

  • N.N. Moiseyev, V.V. Rumyantsev, Dynamic Stability of Bodies Containing Fluid (Springer, New York, 1968)

    CrossRef  Google Scholar 

  • M.V. Nezlin, Negative-energy waves and the anomalous Doppler effect. Sov. Phys. Uspekhi 19, 946–954 (1976)

    CrossRef  Google Scholar 

  • E.L. Nicolai, On the stability of the rectilinear form of equilibrium of a bar in compression and torsion. Izvestia Leningradskogo Politechnicheskogo Instituta 31, 201–231 (1928)

    Google Scholar 

  • E.L. Nicolai, On the problem of the stability of a bar in torsion. Vestnik Mechaniki i Prikladnoi Matematiki 1, 41–58 (1929)

    Google Scholar 

  • O.M. O’Reilly, N.K. Malhotra, N.S. Namachchivaya, Reversible dynamical systems - dissipation-induced destabilization and follower forces. Appl. Math. Comput. 70, 273–282 (1995)

    MathSciNet  MATH  Google Scholar 

  • O.M. O’Reilly, N.K. Malhotra, N.S. Namachchivaya, Some aspects of destabilization in reversible dynamical systems with application to follower forces. Nonlinear Dyn. 10, 63–87 (1996)

    MathSciNet  CrossRef  Google Scholar 

  • L.A. Ostrovskii, S.A. Rybak, L.S. Tsimring, Negative energy waves in hydrodynamics. Sov. Phys. Usp. 29, 1040–1052 (1986)

    CrossRef  Google Scholar 

  • M.P. Païdoussis, Fluid-Structure Interactions, 2nd edn. (Academic Press, Oxford, 2016)

    Google Scholar 

  • D. Phillips, S. Simpson, S. Hanna, Chapter 3 - optomechanical microtools and shape-induced forces, in Light Robotics: Structure-Mediated Nanobiophotonics, ed. by J. Glückstad, D. Palima (Elsevier, Amsterdam, 2017), pp. 65–98

    CrossRef  Google Scholar 

  • L. Pigolotti, C. Mannini, G. Bartoli, Destabilizing effect of damping on the post-critical flutter oscillations of flat plates. Meccanica 52(13), 3149–3164 (2017)

    MathSciNet  CrossRef  Google Scholar 

  • S.M. Ramodanov, V.V. Sidorenko, Dynamics of a rigid body with an ellipsoidal cavity filled with viscous fluid. Int. J. Non-Lin. Mech. 95, 42–46 (2017)

    CrossRef  Google Scholar 

  • P.H. Roberts, K. Stewartson, On the stability of a Maclaurin spheroid with small viscosity. Astrophys. J. 139, 777–790 (1963)

    CrossRef  Google Scholar 

  • A. Rohlmann, T. Zander, M. Rao, G. Bergmann, Applying a follower load delivers realistic results for simulating standing. J. Biomech. 42, 1520–1526 (2009)

    CrossRef  Google Scholar 

  • S. Ryu, Y. Sugiyama, Computational dynamics approach to the effect of damping on stability of a cantilevered column subjected to a follower force. Comput. Struct. 81, 265–271 (2003)

    CrossRef  Google Scholar 

  • S.S. Saw, W.G. Wood, The stability of a damped elastic system with a follower force. J. Mech. Eng. Sci. 17(3), 163–176 (1975)

    CrossRef  Google Scholar 

  • J. Schindler, A. Li, M.C. Zheng, F.M. Ellis, T. Kottos, Experimental study of active LRC circuits with PT symmetries. Phys. Rev. A 84, 040101(R) (2011)

    CrossRef  Google Scholar 

  • A.P. Seyranian, A.A. Mailybaev, Paradox of Nicolai and related effects. Z. angew. Math. Phys. 62, 539–548 (2011)

    MathSciNet  CrossRef  Google Scholar 

  • R.C. Shieh, E.F. Masur, Some general principles of dynamic instability of solid bodies. Z. Angew. Math. Phys. 19, 927–941 (1968)

    CrossRef  Google Scholar 

  • S.H. Simpson, S. Hanna, First-order nonconservative motion of optically trapped nonspherical particles. Phys. Rev. E. 82, 031141 (2010)

    CrossRef  Google Scholar 

  • D.M. Smith, The motion of a rotor carried by a flexible shaft in flexible bearings. Proc. R. Soc. Lond. A 142, 92–118 (1933)

    CrossRef  Google Scholar 

  • K. Stewartson, On the stability of a spinning top containing liquid. J. Fluid Mech. 5, 577–592 (1959)

    MathSciNet  CrossRef  Google Scholar 

  • Y. Sugiyama, K. Kashima, H. Kawagoe, On an unduly simplified model in the non-conservative problems of elastic stability. J. Sound Vib. 45(2), 237–247 (1976)

    CrossRef  Google Scholar 

  • S. Sukhov, A. Dogariu, Non-conservative optical forces. Rep. Prog. Phys. 80, 112001 (2017)

    MathSciNet  CrossRef  Google Scholar 

  • T. Theodorsen, General theory of aerodynamic instability and the mechanism of flutter. Technical Report no. 496. National Advisory Commitee for Aeronautics (NACA) (1935)

    Google Scholar 

  • W. Thomson, On an experimental illustration of minimum energy. Nature 23, 69–70 (1880)

    CrossRef  Google Scholar 

  • W. Thomson, P.G. Tait, Treatise on Natural Philosophy (Cambridge University Press, Cambridge, 1879)

    MATH  Google Scholar 

  • M. Tommasini, O.N. Kirillov, D. Misseroni, D. Bigoni, The destabilizing effect of external damping: singular flutter boundary for the Pflüger column with vanishing external dissipation. J. Mech. Phys. Sol. 91, 204–215 (2016)

    CrossRef  Google Scholar 

  • F.E. Udwadia, Stability of dynamical systems with circulatory forces: generalization of the Merkin theorem. AIAA J. 55(9), 2853–2858 (2017)

    CrossRef  Google Scholar 

  • A.I. Vesnitskii, A.V. Metrikin, Transition radiation in mechanics. Phys.-Uspekhi 39(10), 983–1007 (1996)

    CrossRef  Google Scholar 

  • P. Wu, R. Huang, C. Tischer, A. Jonas, E.-L. Florin, Direct measurement of the nonconservative force field generated by optical tweezers. Phys. Rev. Lett. 103, 108101 (2009)

    CrossRef  Google Scholar 

  • V.A. Yakubovich, V.M. Starzinskii, Linear Differential Equations with Periodic Coefficients, vols. 1 and 2 (Wiley, New York, 1975)

    Google Scholar 

  • R. Zhang, H. Qin, R.C. Davidson, J. Liu, J. Xiao, On the structure of the two-stream instability-complex G-Hamiltonian structure and Krein collisions between positive- and negative-action modes. Physics of Plasmas 23, 072111 (2016)

    CrossRef  Google Scholar 

  • V.F. Zhuravlev, Decomposition of nonlinear generalized forces into potential and circulatory components. Doklady Phys. 52, 339–341 (2007)

    CrossRef  Google Scholar 

  • V.F. Zhuravlev, Analysis of the structure of generalized forces in the Lagrange equations. Mech. Solids 43, 837–842 (2008)

    CrossRef  Google Scholar 

  • H. Ziegler, Stabilitätsprobleme bei geraden Stäben und Wellen. Z. angew. Math. Phys. 2, 265–289 (1951a)

    MathSciNet  CrossRef  Google Scholar 

  • H. Ziegler, Ein nichtkonservatives Stabilitätsproblem. Z. angew. Math. Math. 8(9), 265–266 (1951b)

    Google Scholar 

  • H. Ziegler, Die Stabilitätskriterien der Elastomechanik. Arch. Appl. Mech. 20, 49–56 (1952)

    Google Scholar 

  • H. Ziegler, Linear elastic stability. A critical analysis of methods. First part. ZAMP Z. angew. Math. Phys. 4, 89–121 (1953a)

    CrossRef  Google Scholar 

  • H. Ziegler, Linear elastic stability. A critical analysis of methods, Second part. ZAMP Z. angew. Math. Phys. 4, 167–185 (1953b)

    CrossRef  Google Scholar 

  • H. Ziegler, On the concept of elastic stability. Adv. Appl. Mech. 4, 351–403 (1956)

    Google Scholar 

  • V.I. Zubov, Canonical structure of the vector force field, in Problems of Mechanics of Deformable Solid Bodies – Special issue dedicated to the 60th Birthday of Acad. V. V. Novozhilov (Sudostroenie, Leningrad, 1970), pp. 167–170. [in Russian]

    Google Scholar 

  • O.N. Kirillov, Localizing EP sets in dissipative systems and the self-stability of bicycles. arXiv:1806.03741 (2018)

Download references

Author information

Authors and Affiliations

  1. Northumbria University, Newcastle upon Tyne, NE1 8ST, United Kingdom

    Oleg N. Kirillov

Authors
  1. Oleg N. Kirillov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Oleg N. Kirillov .

Editor information

Editors and Affiliations

  1. Department of Civil, Environmental and Mechanical Engineering, University of Trento, Trento, Italy

    Prof. Dr. Davide Bigoni

  2. Department of Mathematics, Physics and Electrical Engineering, Northumbria University, Newcastle upon Tyne, United Kingdom

    Prof. Dr. Oleg Kirillov

Rights and permissions

Reprints and Permissions

Copyright information

© 2019 CISM International Centre for Mechanical Sciences

About this chapter

Verify currency and authenticity via CrossMark

Cite this chapter

Kirillov, O.N. (2019). Classical Results and Modern Approaches to Nonconservative Stability. In: Bigoni, D., Kirillov, O. (eds) Dynamic Stability and Bifurcation in Nonconservative Mechanics. CISM International Centre for Mechanical Sciences, vol 586. Springer, Cham. https://doi.org/10.1007/978-3-319-93722-9_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-93722-9_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-93721-2

  • Online ISBN: 978-3-319-93722-9

  • eBook Packages: EngineeringEngineering (R0)