A new analytic algorithm called 'continuous piecewise linearization method' (CPLM) is developed to obtain periodic solutions of freely vibrating Duffing-type oscillators. This simple analytic algorithm is based on continuous piecewise linearization of the nonlinear stiffness with respect to displacement and was shown to produce very accurate results for few iterations. The algorithm is valid for Duffing-type oscillators possessing strong nonlinearity and/or undergoing large-amplitude oscillations. Studies conducted on Duffing oscillators with cubic, cubic-quintic and trigonometric sine stiffness nonlinearities showed that the CPLM results match standard numerical solutions and is more accurate than the popular energy balance method (EBM). Additionally, the present analysis shows that the CPLM is capable of predicting the quasi-linear behaviour observed in the oscillation history of Duffing-type oscillators with strongly nonlinearity and/or large-amplitude oscillations. This quasi-linear behaviour cannot be predicted by the EBM to which the CPLM is compared.
This paper presents a new contact model for analysis of post-yield indentation of a halfspace target by a spherical indenter. Unlike other existing models, the elastoplastic regime of the present model was modelled using two distinct force-indentation relationships based on experimentally and theoretically established indentation characteristics of the elastoplastic regime. The constants in the model were derived from continuity conditions and indention theory. Simulations of the present model show good prediction of experimental data. Also, an approach for determining the maximum contact force and indentation of an elastoplastic half-space from the impact conditions has been proposed.
The governing equation of a half-space impact is generally nonlinear and it is normally solved using numerical techniques that are mostly conditionally stable and require many iteration steps for convergence of the solution. In this paper, we present the forceindentation linearisation method (FILM), an approximate technique that produces closedform solutions of piecewise linearisation of the governing nonlinear differential equation and is capable of producing accurate impact response for an elastoplastic half-space impact.In contrast to the existing numerical techniques, which discretise the impact force or variable of interest in the time-domain, the present technique discretises the impact force with respect to the indentation using successive piecewise linear approximations.Generalised closed-form solutions were derived for each piecewise approximation, and this was used to develop an iterative algorithm for updating the solutions from one piecewise approximation to the next. The results of the present technique matched with results obtained by direct numerical integration of the governing nonlinear differential equation for 1 Corresponding author: Systems, Power and Energy Research Division, School of Engineering, University of Glasgow, G12 8QQ, Glasgow Scotland, UK. Email: a.big-alabo.1@research.gla.ac.uk 2 a half-space impact, and the FILM was found to converge to the results of the numerical solution after a few iterations; typically between five and ten iterations. The FILM is simple, inherently stable, converges quickly, gives accurate results, and it can be implemented manually; these features makes it potentially more attractive than the comparable numerical methods.
A new cubication method is proposed for periodic solution of nonlinear Hamiltonian oscillators. The method is formulated based on quasi-static equilibrium of the original oscillator and the undamped cubic Duffing oscillator. The cubication constants derived from the present cubication method are always based on elementary functions and are simpler than the constants derived by other cubication methods. The present method was verified using three common examples of strongly nonlinear oscillators and was found to give reasonably accurate results. The method can be used to introduce nonlinear oscillators in relevant undergraduate physics and mechanics courses.
This paper presents approximate periodic solutions to the anharmonic (i.e. not harmonic or non-sinusoidal) response of a simple pendulum undergoing moderate- to large-amplitude oscillations. The approximate solutions were derived by using a modified continuous piecewise linearization method that enabled very accurate solutions to the pendulum oscillations for the entire range of possible amplitudes i.e. [Formula: see text]. The present solution method is very simple and can be used to obtain amplitude-frequency solutions as well as the displacement and velocity histories of the simple pendulum without the need for a complementary method. The purpose of this paper is to present simple and accurate approximate analytical solutions to the large-amplitude oscillations of the simple pendulum that can be applied by undergraduates.
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