# Reply to Comment on 'Nonlinear dynamics of a position-dependent mass-driven Duffing-type oscillator'

TL;DR: In this article, an extended Lagrange equation involving a non-conservative force term when the particle mass varies with position has been proposed, which yields precisely the form of Newton's equation expected for a particle possessing a position-dependent mass.

Abstract: In response to the comment of Mustafa (2013 J. Phys. A: Math. Theor. 46 368001) we justify our stand of considering an extended Lagrange equation involving a non-conservative force term when the particle mass varies with position. As has been known for some time, such an extended form not only takes into account the principle of virtual work applied to D’Alembert’s principle but yields precisely the form of Newton’s equation expected for a particle possessing a position-dependent mass.

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TL;DR: In this article, a nonlinear dynamics study of a RLC series circuit modeled by a generalized Van der Pol oscillator is investigated, where the external excitation strength, harmonic, subharmonic and superharmonic oscillatory states are obtained using the multiple time scales method.

Abstract: Abstract In this paper, nonlinear dynamics study of a RLC series circuit modeled by a generalized Van der Pol oscillator is investigated. After establishing a new general class of nonlinear ordinary differential equation, a forced Van der Pol oscillator subjected to an inertial nonlinearity is derived. According to the external excitation strength, harmonic, subharmonic and superharmonic oscillatory states are obtained using the multiple time scales method. Bifurcation diagrams displayed by the model for each system parameter are performed numerically through the fourth-order Runge–Kutta algorithm.

8 citations

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10 Oct 2014

TL;DR: In this paper, the phase space trajectories for position-dependent mass oscillator, Scarf and Poschl-Teller potentials with Gaussian and singular masses are presented.

Abstract: In this work the dynamical equations for a system with position-dependent mass are considered. The phase space trajectories are constructed by means of the factorization method for classical systems. To illustrate how this formalism works the phase space trajectories for position-dependent mass oscillator, Scarf and Poschl-Teller potentials with Gaussian and singular masses are presented.

4 citations

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CINVESTAV

^{1}TL;DR: The present work is an extended abstract from a series of lectures addressed to introduce elements of the theory of position-dependent mass systems in both, classical and quantum mechanics as mentioned in this paper, which is an extension of a previous work.

Abstract: The present work is an extended abstract from a series of lectures addressed to introduce elements of the theory of position-dependent mass systems in both, classical and quantum mechanics.

3 citations

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01 May 2022

TL;DR: In this article , a position-dependent mass (PDM) oscillator with doubly singular mass distribution function describing the vibrational inversion mode of the NH molecule was examined.

Abstract: We examine vibrational resonance (VR) in a position-dependent mass (PDM) oscillator with doubly singular mass distribution function describing the vibrational inversion mode of NH $$_{3}$$ molecule. The impacts of the PDM parameters $$(m_{0}, a,\eta )$$ on VR were studied by computing the response amplitudes as functions of the amplitude of high-frequency component of the dual-frequency driving forces and the PDM parameters. We show for the first time that, beside the significant roles played by the parameters of the variable mass in inducing and controlling resonances similar to the forcing parameters, the variable mass parameters impact on the resonance characteristics by leading the system from single resonance into double resonance.

3 citations

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TL;DR: The derivation and application of the Lagrange equations of motion to systems with mass varying explicitly with position are discussed in this paper, where the concept of added mass tensor is introduced.

Abstract: The derivation and application of the Lagrange equations of motion to systems with mass varying explicitly with position are discussed Two perspectives can be followed: systems with a material type of source, attached to particles continuously gaining or losing mass, and systems for which the variation of mass is of a nonlinear control volume type, mass trespassing a control surface This is the case if for some theoretical or practical reason, a partition into subsystems is considered An important class of problems in which the extended Lagrange equations turn to be useful emerges from hydromechanics, whenever a finite number of generalized coordinates can be used, under the concept of the added mass tensor A particular and interesting one is addressed in the present paper: the classical hydrodynamic impact of a rigid body against a liquid free surface

77 citations

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TL;DR: In this paper, a nonlinear vibrating machine and a rotor with variable mass are considered, and Noether's identity for such a case is expanded by the terms that describe the mass variation.

Abstract: It is based on Noether's theorem to the existence of conservation laws and D'Alembert variational principle. In general case, a dynamic system with variable mass is purely nonconservative. Noether's identity for such a case is expanded by the terms that describe the mass variation. If Noether's identity is satisfied, a conservation law exist. Two groups of systems are considered: a nonlinear vibrating machine and a rotor with variable mass

42 citations

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TL;DR: In this paper, a naive application, without any special consideration on non-conservative generalized forces, leads to equations of motions which lack (or exceed) terms of the form 1/2(m/q) where q is a generalized coordinate.

Abstract: The usual Lagrange equations of motion cannot be directly applied to systems with mass varying explicitly with position. In this particular context, a naive application, without any special consideration on non-conservative generalized forces, leads to equations of motions which lack (or exceed) terms of the form 1/2(¶m/¶q.2), where q is a generalized coordinate. This paper intends to discuss the issue a little further, by treating some applications in offshore engineering under the analytic mechanics point of view.

23 citations

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TL;DR: In this paper, the equivalence between Euler-Lagrange's and Newton's equations is proved and documented through the proper invertible coordinate transformation and the introduction of a new PDM byproducted reaction-type force.

Abstract: Using a generalized coordinate along with a proper invertible coordinate transformation, we show that the Euler-Lagrange equation used by Bagchi et al. 16 is in clear violation of the Hamilton's principle. We also show that Newton's equation of motion they have used is not in a form that satisfies the dynamics of position-dependent mass (PDM) settings.. The equivalence between Euler-Lagrange's and Newton's equations is now proved and documented through the proper invertible coordinate transformation and the introduction of a new PDM byproducted reaction-type force. The total mechanical energy for the PDM is shown to be conservative (i.e., dE/dt=0, unlike Bagchi et al.'s 16 observation).

22 citations

### "Reply to Comment on 'Nonlinear dyna..." refers result in this paper

...As also agreed in [11], even with his proposed amendment of Newton’s law, the numerical results of our study would not be affected....

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