Dynamic analyses of plane frames by integral equations for bars and Timoshenko beams

In this paper, the Analog Equation Method (AEM), a BEM-based method, is employed to the nonlinear analysis of a Bernoulli-Euler beam with variable stiffness undergoing large deflections, under general boundary conditions which maybe nonlinear. As the cross-sectional properties of the beam vary along its axis, the coefficients of the differential equations governing the equilibrium of the beam are variable. The formulation is in terms of the displacements. The governing equations are derived in both deformed and undeformed configuration and the deviations of the two approaches are studied. Using the concept of the analog equation, the two coupled nonlinear differential equations with variable coefficients are replaced by two uncoupled linear ones pertaining to the axial and transverse deformation of a substitute beam with unit axial and bending stiffness, respectively, under fictitious load distributions. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. Several beams are analyzed under various boundary conditions and loadings to illustrate the merits of the method as well as its applicability, efficiency and accuracy.

Large deflection analysis of beams with variable stiffness

An Euler–Bernoulli-like finite element method for Timoshenko beams

This article presents a novel six degrees-of-freedom nanopositioning stage having a low-profile shape. The six degrees-of-freedom motion is achieved in a parallel configuration comprising four crab-leg flexure guide mechanisms and eight electromagnetic actuators. The crab-leg flexure can be engineered to exhibit similar stiffness levels in all the directions. The noncontact direct-drive nature of the electromagnetic actuators and the symmetric layout simplify the kinematics and result in considerably low coupling between the motion axes. For design optimization, we derived an analytical model that described the static and dynamic performances of the proposed stage using the matrix method. Then, we fabricated the stage with a stroke of 0.5 mm in the translational directions and 5 mrad in the rotational directions. The stage measured 250 mm × 250 mm × 57.4 mm. In an open-loop operation, the six-axis motions exhibited linear and repeatable performances. The maximum parasitic error was less than 1.8% of full scale. The tracking errors for the closed-loop control of the 5 Hz sinusoidal trajectory were less than 222 nm and 4.2  μ rad. The temperature elevation was 5.8 °C after 3 h using half of the entire power consumption. The stage was utilized for developing an autofocusing system in dynamic scanning using an optical microscope.

Six Degrees-of-Freedom Direct-Driven Nanopositioning Stage Using Crab-Leg Flexures

Bolted joints often have a significant effect on the dynamical behavior of assembled mechanical structures. An accurate model of an assembled structure depends on correctly determining and identifying the dynamic parameters of bolted joints. This paper presents an alternative method for identifying these dynamic parameters using structure’s natural frequency and damping. A novel experiment is designed with a test piece consisting of only bolted joints, with the governing equations of the test piece established using the analytical method. The relationships between the equivalent dynamic parameters of the bolted joints and the natural frequencies and damping ratios of the test piece are determined for both the normal and tangential directions. The parameter identification problem for bolted joints is thus transformed into a test of the natural frequency and the damping ratio of the test piece. In order to check the accuracy of the proposed identification method, the test piece and bolted joints are modeled using the finite element method (FEM) and the dynamic properties of the test piece are analyzed. The maximum error between the natural frequencies of the FEM result and the experimental values in the normal and tangential models are 4.73% and 0.34%, respectively. The result indicates that the proposed method is valid for the dynamic parameter identification of bolted joints.

H. Antes

Papers

Fundamental solution and integral equations for Timoshenko beams