A Galerkin–Newton Algorithm for Solution of a Kirchhoff-Type Static Equation

In this paper, an algorithm is proposed to find an approximate solution for the Kirchhoff -type nonlinear differential equation, which describes the static state of a beam. The solution of the problem consists of two parts. First, we apply the Galerkin method. Next, to solve the obtained discrete system of equations, we use the Newton iteration method. The algorithm total error is estimated. The results of the numerical experiment are given.

Statics and Dynamics of V-shaped Micro-beams Under Axial Forces

We present an investigation into the static and dynamic behaviors of electrostatically actuated in-plane micro-electro-mechanical V-shaped micro-beam under axial loads. The micro-beams are actuated with two separate electrodes of uniform air-gap across their length. The effects of the initial rise and DC bias voltage are examined while varying the axial loads ranging from compressive to tensile. The numerical analysis is based on a nonlinear equation of motion of a shallow V-shaped micro-beam. The static equation is solved using a reduced-order model based on the Galerkin procedure. Then, the eigenvalue problem of the structure is solved for various equilibrium positions. The analytical model is validated by comparing to an experimental case study. The results show rich and diverse static and dynamic behavior. It is shown that the micro-beam may exhibit only pull-in or snap-through and pull-in instabilities. Various multi-state and hysterics behaviors are demonstrated when varying the actuation forces and the initial rise. High tunability is demonstrated when varying the axial and DC loads for the first two symmetric vibration modes. Such rich behavior can be very useful for high performance micro-scale applications designs.

A New Index of Static Actuation Force Sensitivity of Mechanisms Based on Unified Forward Jacobian Matrix

This paper proposes a novel performance index, which is called static actuation force sensitivity (SAFS), to investigate the response of the actuation forces when the amplitude of the suffered load of the end-effector has a change. Smaller SAFS can protect the actuations, and the load is mainly suffered by the structural constraints. This work starts with the construction of the unified forward Jacobian matrix of both serial and parallel mechanisms by screw theory. Then, with the forward Jacobian matrix, the inverse static equation is established. SAFS is thus introduced by the “partial differential” operation on the inverse static equation. SAFS is only related to the position of the whole mechanism and the direction of the suffered load, but not related to the detailed value of the amplitude of the load and the detailed value of the actuation forces; thus, SAFS can reveal the essence of static force capacities of the mechanisms. The example mechanism (namely, the 3revolute-prismatic-spherical (RPS) parallel mechanism) is used to illustrate the distribution of SAFS both over the workspace and at a certain pose. The analysis method of SAFS and the proposed index are expected to be applied to the pose optimization in the motion planning of the mechanisms to protect the actuations.

Mechanical properties of hexagonal boron nitride monolayers: Finite element and analytical predictions

The mechanical response of two-dimensional nanostructures may be significantly affected by their size. In this work, a molecular structural mechanics model is developed and is implemented in order to predict the nanomechanical behavior and calculate the corresponding elastic properties of hexagonal boron nitride sheets and describe their size-dependence. The finite element approach utilizes appropriate spring-like elements for the modeling of interactions between atoms within the hexagonal boron nitride structure, the stiffness constants of which are obtained by the molecular mechanics theory. Adopting conventional finite element techniques, the global stiffness matrix of the structure of a desired sheet size can be assembled. Applying appropriate boundary conditions, the governing equilibrium static equation can be solved and the elastic mechanical properties as Young’s modulus, shear modulus, and Poisson’s ratio of the structure can be calculated. Fitting the results of the mechanical properties calculated by the finite element analysis, analytical–empirical equations are proposed for their direct prediction for an hexagonal boron nitride sheet having the size parameters of the structure as independent variables.

Determinants of the Size of the Government Expenditure: An Empirical Study on Bangladesh

Analysis of the nature of government expenditure constitutes a central concern in economic literature. This is because many countries of the world consistently have increased the size of government expenditure. Bangladesh has done the same practice over the last few decades. There is a need to investigate the factors which determine the size of the public spending of Bangladesh. The error correction modeling technique for the short-run dynamic equation and ordinary least square (OLS) for long-run static equation are used over the period 1973 to 2016 to this purpose.  The results of the short run dynamic equation and long-run static equation showed that external debt, real GDP, urbanization, tax, and non-tax government revenue positively influence the government expenditure where dependency on foreign aid and trade openness adversely affect it. The study recommends that the government should take proper step to expand the revenue base, stimulate the economic growth and reduce the external debt, and foreign aid.

A numerical method for calculating the misalignments of planetary gears

Because misalignments derived from the deflections of transmission systems have significant effects on the load capacity of planetary gears, these misalignments should be accurately considered in the analysis of planetary gears. Here, we develop a new approach for misalignment calculations of cylindrical planetary gears. A nonlinear model of a planetary gear transmission system is built based on the finite element method and nonlinear bearing theory for misalignment calculations that can precisely simulate the structural characteristics and mechanical properties of a planetary gear system. The nonlinear static equation of a planetary system is solved efficiently using the Newton–Raphson method. Gear misalignments of all the planet branches are determined by the results of the system static analysis. The reliability and advantages of the proposed method are discussed via case studies. The effects of including the variation of the planet positions and the nonlinearity of the bearing stiffness on the planetary gear misalignments under different load conditions are studied. The misalignments can be reliably determined using the proposed method for calculating the load capacity of planetary gears.

Statics and Dynamics of MEMS Arches Under Axial Forces

This works aims to investigate the effect of axial forces on the static behavior and the fundamental natural frequency of electrostatically actuated MEMS arches. The analysis is based on a nonlinear equation of motion of a shallow arch under axial and electrostatic forces. The static equation is solved using a reduced-order model based on the Galerkin procedure. The effects of the axial and electrostatic forces on the static response are examined. Then, the eigenvalue problem of the arch is solved for various equilibrium positions. Several results are shown for the variations of the natural frequency and equilibrium position of the arch under axial forces ranging from compressive loads beyond buckling to tensile loads and for voltage loads starting from small values to large values near the pull-in instability. It is found that the dynamics of MEMS arches are very sensitive to axial forces, which may be induced unintentionally through microfabrication processes or due to temperature variations while in use. On the other hand, it is shown that axial forces can be used deliberately to control the dynamics of MEMS arches to achieve desirable functions, such as extending their stable operation range and tuning their natural frequencies.