The conservative Helmholtz-Duffing oscillator is analyzed by means of three analytical techniques. The max-min, second-order of the Hamiltonian, and the global error minimization approaches are applied to achieve natural frequencies. The obtained results are compared with the homotopy perturbation method and numerical solutions. The results show that second-order of the global error minimization method is very accurate, so it can be widely applicable in engineering problems.

In this article, the 3D printed polyamide12/carbon nanotube (PA12/CNT) nanocomposites were joined using laser welding. The factorial design method was applied to study the effect of laser power, welding velocity, and CNT content on the tensile and impact strengths of welded samples. The differential scanning calorimetry and thermogravimetric analysis were also used to measure the thermal stability and weight loss temperature of the nanocomposite. The results indicated that the increase of CNTs enhanced the thermal stability of the nanocomposite and so increased its melting point and crystallinity percentage. The impact and tensile strengths of the weld enhanced when laser power elevated from 40 to 50 W, but elevation of laser power up to 60 W decreased the impact and tensile strengths due to thermal degradation of the nanocomposite. A rise in welding velocity from 10 to 20 mm/s enhanced the tensile strength, while the impact strength initially enhanced and then decreased. The highest tensile strength obtained at 2 wt% CNT, whereas the addition of CNTs up to 4 wt% led to their agglomeration, which resulted in the reduction of impact and tensile strengths. Finally, the optimal conditions for concurrent increase of impact and tensile strengths include the laser power of 51 W, welding velocity of 12.5 mm/s, and CNT content of 2.3 wt%.