newton-raphson method

The free vibration features of rotating functionally graded microbeams in thermal environmentare presented in this paper. The governing equations are extracted on the basis of the Euler-Bernoulli beam assumptions beside the modified couple stress theory. The finite elementmethod is applied on the weak form of the strain and the kinetic energies to extract the naturalfrequencies and the associated modeshapes. The nonlinear static equations of motion due to therotation and the thermal environment are treated employing the Newton-Raphson technique.Moreover, the natural frequencies are estimated from the linearized equations of motion aboutthe static configuration. After the validation of the present results, the rotation speed, thematerial length scale parameter, the temperature change, the power law exponent and theslenderness ratio impacts on the fundamental natural frequency and the first and the secondmodeshapes are examined. The outcomes indicate the increment of the natural frequency aftera threshold value of the power law exponent depends on a given rotation speed for the rotatingmicrobeams in comparison with the stationary microbeams. Furthermore, the modeshapes ofrotating functionally graded microbeams vary by the power law exponent while for stationaryfunctionally graded microbeams the modeshapes are invariant with respect to the power lawexponent even in the presence of the thermal environment.

In this paper, the structure of a novel 6 DOF active joint using cable driven parallel mechanisms is designed, analyzed and presented. This joint consists of two fixed and mobile frames, and unlike all the commonly used cable driven parallel mechanisms, for first time the outer frame is mobile and the interior frame is considered to be fixed. The mobile frame is attached to the fixed frame by the 8 cables and moves through the tensioning and lengthening of the cables. This new structure can be applied to the wrist mechanism, motion simulators, power balances, haptic interfaces, and Etc. Since the parallel mechanism is a closed loop system and its equations cannot be separated regards to the Cartesian space variables, forward kinematics cannot be solved easily and analytically. Several methods have been proposed to solve the forward kinetic of parallel mechanisms, including numerical optimization and methods, in which the Newton-Ruffson numerical method is used here. Due to the unilateral nature of cables that are only effective in pulling, cables must always be tension. For this purpose, a tension optimization algorithm is presented and, by solving this algorithm for all possible positions of the mobile frame, the workspace of the mechanism is obtained. The results show that the Newton-Raphson method has an appropriate convergence rate and the tension algorithm is capable of determining the tension forces of the cables in the desired range.

Carbon nanotubes have an important role in reinforcing nanocomposits. Many experimental observations have shown that in the most nanostructures such as nanocomposites, carbon nanotubes (CNTs) are often characterized by a certain degree of waviness along their axial direction. In the present paper, the effects of initial curvature, influence of surrounding medium that is modeled as Winkler elastic foundation on behavior of slightly curved carbon nanotubes are investigated. To capture the small size effects, nonlocal elasticity theory is implemented. Bending governing equations are derived using the principle of minimum total potential energy and these nonlinear equations are solved by Newton Raphson method. It is shown that the larger the initial curvature, the higher deflection can occur. Furthermore, neglecting the effect of initial curvature of CNTs can lead to incorrect results.