Parallelogram

Abstract: SUMMARY A new algorithm is developed to improve the accuracy and efficiency of the material point method for problems involving extremely large tensile deformations and rotations. In the proposed procedure, particle domains are convected with the material motion more accurately than in the generalized interpolation material point method. This feature is crucial to eliminate instability in extension, which is a common shortcoming of most particle methods. Also, a novel alternative set of grid basis functions is proposed for efficiently calculating nodal force and consistent mass integrals on the grid. Specifically, by taking advantage of initially parallelogram-shaped particle domains, and treating the deformation gradient as constant over the particle domain, the convected particle domain is a reshaped parallelogram in the deformed configuration. Accordingly, an alternative grid basis function over the particle domain is constructed by a standard 4-node finite element interpolation on the parallelogram. Effectiveness of the proposed modifications is demonstrated using several large deformation solid mechanics problems. Copyright 2011 John Wiley & Sons, Ltd.

Abstract: A fast maximum likelihood algorithm is presented that jointly estimates the frequency and frequency rate of a sinusoid corrupted by additive Gaussian white noise It consists of a coarse search and a fine search First the two-dimensional frequency-frequency rate plane is subdivided into parallelograms whose size depends on the region of convergence of Newton's method used in maximizing the log-likelihood function (LLF) The size of the parallelogram is explicitly computed and is optimal for the method used The coarse search consists of maximizing the LLF over the vertices of the parallelograms Then starting at the vertex where the LLF attained its maximum, a two-dimensional Newton's method to find the absolute maximum of the LLF is implemented This last step consists of the fine search The rate of convergence of Newton's method is cubic, and is extremely fast Furthermore Newton's method will converge after two iterations when the starting point used in the method lies within 75 percent of the distances defined by the parallelogram of convergence whose center coincides with the true values of frequency and frequency rate In this case, the root mean square error (RMSEs) for frequency and frequency rate are practically equal to the Cramer-Rao bound at all signal-to-noise ratio (SNR)?15 dB The frequency-frequency rate ambiguity function is shown to be even and its periodicities are extracted

Abstract: This paper presents the design analysis fabrication and testing of a high-bandwidth piezo-driven parallel kinematic nanopositioning XY stage. The monolithic stage design has two axes and each axis is composed of a doubly clamped beam and a parallelogram hybrid flexure with compliant beams and circular flexure hinges. The doubly clamped beam that is actuated by a piezoelectric actuator acts as a linear prismatic axis. The parallelogram hybrid flexures are used to decouple the actuation effect from the other axis. The mechanism design decouples the motion in the X- and Y-directions and restricts parasitic rotations in the XY plane while allowing for an increased bandwidth with linear kinematics in the operating region. Kinematic and dynamic analysis shows that the mechanical structure of the stage has decoupled motion in XY-direction while achieving high bandwidth and good linearity. The stage is actuated by piezoelectric stack actuators, and two capacitive gauges were added to the system to build a closed-loop positioning system. The results from frequency tests show that the resonant frequencies of the two vibrational modes are over 8 kHz. The stage is capable of about 15 μm of motion along each axis with a resolution of about 1 nm. Due to parallel kinematic mechanism design, a uniform performance is achieved across the workspace. A PI controller is implemented for the stage and a closed-loop bandwidth of 2 kHz is obtained.