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abscissa approximate solution assumed calculate the amplitude Coli consists of finding curve damping degree of freedom distance Duffing energy dissipation energy method gives equation 9a equilibrium exact solution external force find a periodic forced vibration Fourier series gmco2a3 gmu2 graphical construction graphical solution greater accuracy HenI inertia force intersection kind of spring kinetic energy line of slope linear spring Martienssen solution Martienssen's solution mw2ai mw2x2max NON-HARMONIC non-linear spring numerical onbe ordinate intercept perature periodic solution problem consists problem of forced quarter cycle quarter wave energy requirement right hand member satisfied shown in Fig sine wave sinusoidal motion solution exists SOLUTION OF MARTIENSSEN solutions are given spring characteristic spring force straight line superposition term approximation third harmonic three forces tion two-term approximation undamped single degree vertical axis wave energy method xmax