New Modification on Heun’s Method Based on Contraharmonic Mean for Solving Initial Value Problems with High Efficiency

In this paper, small modification on Improved Euler’s method (Heun’s method) is proposed to improve the efficiency so as to solve ordinary differential equations with initial condition by assuming the tangent slope as an average of the arithmetic mean and contra-harmonic mean. In order to validate the conclusion, the stability, consistency, and accuracy of the system were evaluated and numerical results were presented, and it was recognized that the proposed method is more stable, consistent, and accurate with high performance.

Compensation of Hysteresis in the Piezoelectric Nanopositioning Stage under Reciprocating Linear Voltage Based on a Mark-Segmented PI Model

The nanopositioning stage with a piezoelectric driver usually compensates for the nonlinear outer-loop hysteresis characteristic of the piezoelectric effect using the Prandtl–Ishlinskii (PI) model under a single-ring linear voltage, but cannot accurately describe the characteristics of the inner-loop hysteresis under the reciprocating linear voltage. In order to improve the accuracy of the nanopositioning, this study designs a nanopositioning stage with a double-parallel guiding mechanism. On the basis of the classical PI model, the study firstly identifies the hysteresis rate tangent slope mark points, then segments and finally proposes a phenomenological model—the mark-segmented Prandtl–Ishlinskii (MSPI) model. The MSPI model, which is fitted together by each segment, can further improve the fitting accuracy of the outer-loop hysteresis nonlinearity, while describing the inner-loop hysteresis nonlinearity perfectly. The experimental results of the inverse model compensation control show that the MSPI model can achieve 99.6% reciprocating linear voltage inner-loop characteristic accuracy. Compared with the classical PI model, the 81.6% accuracy of the hysteresis loop outer loop is improved.

Analytical Solution of Thick Piezoelectric Curved Beams with Variable Curvature Considering Shearing Deformation

In this paper, an analytical method based on Timoshenko theory is derived for obtaining the in-plane static closed-form general solutions of deep curved laminated piezoelectric beams with variable curvatures. The equivalent modulus of elasticity is utilized to take into account the material couplings in the laminated beam. The linear piezoelectric effect is considered to develop the static governing equations. The governing differential equations are formulated as functions of the angle of tangent slope by introducing the coordinate system defined by the arc length of the centroidal axis and the angle of tangent slope. To solve the governing equations, defined are the fundamental geometric properties, such as the moments of the arc length with respect to horizontal and vertical axes. As the radius is known, the fundamental geometric quantities can be calculated to obtain the static closed-form solutions of the axial force, shear force, bending moment, rotation angle, and displacement fields at any cross-section of curved beams. The closed-form solutions of the circle beams covered with piezoelectric layers under various loading cases are presented. The results show the consistency in comparison with finite results. Solutions of the non-dimensional displacements for the laminated circular and spiral curved beams with different lay-ups are available. The non-dimensional displacements with geometry and material parameters are also investigated.

Analysis of finite deformation of curved beams bonded with piezoelectric actuating layers

Piezoelectric laminated curved beams or laminated curved smart beams, one of the most popular elements, are widely used in nano- or micro-electromechanical systems due to their excellent properties such as small volume, lightweight, and quick response. In this article, the finite deformation of piezoelectric laminated curved beams is analyzed based on Lagrangian and Eulerian description. The piezoelectric actuating character for the deflection in the curved beams bonded with piezoelectric film (polyvinylidene fluoride) driving layers is investigated. Choosing the deformed radius of curvature and tangent slope angle as fundamental parameters, the governing equations of laminated curved smart beams under static mechanical and electrical loadings are derived. First, the equilibrium equations are deduced and decoupled using the deformed angle of tangent slope as the only variant. And then the analytical solutions of laminated curved smart beams are presented using harmonic functions. Finally, the static deformations of the laminated curved smart beams are calculated by this method and the finite element method. The results exhibit good consistency and show the validation of the present method. Circular and spiral beams covered with piezoelectric layers are researched further. Effects of radius, thickness ratios, and stacking sequence on deflections of the piezoelectric laminated beams are explored as well.

Stability and Large Deformation of 2-D Laminate Circular Thin Curved Beams

In this research, both un-deformed or Lagrangian state and deformed or Eulerian state are used to derive for stability analysis and large deformation. By choosing the deformed radius of curvature and deformed angle of tangent slope as parameters, the governing equations of laminated curved beam under static loading are transformed into a set of equations in terms of angle of tangent slope. All the quantities of axial force, shear force, radial and tangential displacements of circular thin curved beam are expressed as functions of angle of tangent slope by using laminate theory. The buckling load and large deformation analytical solutions of circular thin curved beam under a pair of forces are presented as well.

Novel Concentration-Killing Curve Method for Estimation of Bactericidal Potency of Antibiotics in an In Vitro Dynamic Model

ABSTRACT The bactericidal pharmacodynamics of antibiotics against Escherichia coli were analyzed by a concentration-killing curve (CKC) approach, and the novel parameters median bactericidal concentration (BC50) and bactericidal intensity (r) for bactericidal potency were proposed. By using the agar plate method, about 500 E. coli cells were inoculated onto Luria-Bertani plates containing a series of antibiotic concentrations, and after 24 h of incubation at 37°C, all the viable colonies were enumerated. This resulted in a sigmoidal CKC that could be perfectly fitted (R 2 > 0.9) with the function N = N 0/[1 + e r(x − BC50)], where N is number of colonies surviving on each plate with an x series of concentrations of an antibiotic, and N 0 represents the meaningful inoculum size. Construction of the CKC method was based on the bactericidal effect of each antibiotic against the bacterial strain versus the concentration in two dimensions and may be a more valid, accurate, and reproducible method for estimating the bactericidal effect than the endpoint minimum bactericidal concentration (MBC) method. Mathematically, the CKC approach was point symmetrical toward its inflexion (BC50, N 0/2); thus, 2BC50 could replace MBC. The parameter BC1 can be defined as BC50 + [ln(N 0 − 1)/r], which is the drug concentration at which only one colony survived and which is the least critical value of MBC in the CKC. The variate r, which determined the tangent slope on inflexion when N 0 was limited, could estimate the bactericidal intensity of an antibiotic. This verified that the CKC approach may be useful in studies with other classes of antibiotics and has considerable value as a tool for the accurate and proper administration of antibiotics.