zbMATH — the first resource for mathematics

Analytical solution of the damped Helmholtz-Duffing equation. (English) Zbl 1259.34004
Consider the differential equation \[ \ddot x+ Ax+ 2\nu\dot x+ Bx^2+\varepsilon x^3= 0.\tag{\(*\)} \] Under the assumption \(A= {3B^2+ 8\varepsilon\nu\over 9\varepsilon}\), the author derives an analytic solution of equation \((*)\) by using Jacobian elliptic functions.

34A05 Explicit solutions, first integrals of ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems, general theory
Full Text: DOI
[1] Han, W.; Petyt, M., Geometrically nonlinear vibration analysis of thin, rectangular plates using the hierarchical finite element method: 1st mode of laminated plates and higher modes of isotropic and laminated plates, Comput struct., 63, 2, 295-308, (1997) · Zbl 0933.74535
[2] Lai, S.K.; Harrington, J.; Xiang, Y.; Chow, K.W., Accurate analytical perturbation approach for large amplitude vibration of functionally graded beams, Int. J. non-linear mech., (2011)
[3] Chen, Y.Z.; Lin, X.Y., Several numerical solution techniques for nonlinear eardrum-type oscillations, J sound vib., 296, 1059-1067, (2006)
[4] Bonisoli, E.; Vigliani, A., Passive elasto-magnetic suspensions: nonlinear models and experimental outcomes, Mech. res. comm., 34, 385-394, (2007)
[5] Elías-Zúñiga, A., Exact solution of the quadratic mixed-parity helmholtz – duffing oscillator, Appl. math. comput., 218, 7590-7594, (2012) · Zbl 1245.34002
[6] Feng, Z., Monotonous property of non-oscillations of the damped duffing’s equation, Chaos solitons fractals, 28, 463-471, (2006) · Zbl 1097.34026
[7] Chandrasekar, V.K.; Senthilvelan, M.; Lakshmanam, M., New aspects of integrability of force-free Duffing-van der Pol oscillator and related nonlinear systems, J. phys. A: math. gen., 37, 4527-4534, (2004) · Zbl 1069.34055
[8] Almendral, J.A.; Sanjuán, M.A.F., Integrability and symmetries for the Helmholtz oscillator with friction, J. phys. A: math. gen., 36, 695-710, (2003) · Zbl 1066.70015
[9] Compeán, F.I.; Olvera, D.; Campa, F.J.; López de Lacalle, L.N.; Elías-Zúñiga, A.; Rodríguez, C.A., Characterization and stability analysis of a multivariable milling tool by the enhanced multistage homotopy perturbation method, Int. J. Mach. tool manu., 57, 27-33, (2012)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.