Analysis of the nonlinear vibration of a two-mass–spring system with linear and nonlinear stiffness

An analytical approach is developed for areas of nonlinear science such as the nonlinear free vibration of a conservative, two-degree-of-freedom mass–spring system having linear and nonlinear stiffnesses. The main contribution of this research is twofold. First, it introduces the transformation of two nonlinear differential equations for a two-mass system using suitable intermediate variables into a single nonlinear differential equation and, more significantly, the treatment of a nonlinear differential system by linearization coupled with Newton’s method. Secondly, the major section is the solving of the governing nonlinear differential equation where the displacement of the two-mass system can be obtained directly from the linear second-order differential equation using a first-order variational approach. The aforementioned approach proposed by J.H. He, who actually developed the method, is exactly He’s variational method. This approach is an explicit method with high validity for resolving strong nonlinear oscillation system problems. Two examples of nonlinear two-degree-of-freedom mass–spring systems are analyzed, and verified with published results and exact solutions. The method can be easily extended to other nonlinear oscillations and so could be widely applicable in engineering and science.     

Preconditioned Iterative Methods for Scattered Data Interpolation

The scattered data interpolation problem in two space dimensions is formulated as a partial differential equation with interpolating side conditions. The system is discretized by the Morley finite element space. The focus of this paper is to study preconditioned iterative methods for the corresponding discrete systems. We introduce block diagonal preconditioners, where a multigrid operator is used for the differential equation part of the system, while we propose an operator constructed from thin plate radial basis functions for the equations corresponding to the interpolation conditions. The effect of the preconditioners are documented by numerical experiments.     

The bending-rotation spectrum of fulminic acid and deuterofulminic acid

A quasilinear molecular model is needed to account for the infrared absorption spectrum of HCNO and DCNO in the spectral region from 100 cm^?1 to 1000 cm^?1. The observed systems of infrared bands arising from the ν₅ vibrational manifold have all been assigned. The rotational structure of the absorption bands at 225 cm^?1, 275 cm^?1, 315 cm^?1, and 317 cm^?1 for HCNO has been resolved using a Fourier spectrometer. The rotational constants and the band centers have been determined for the above bands, which represent the transitions(0000⁰1¹)_c←0000⁰0⁰(0000⁰2²)_c,d←(0000⁰1¹)_c,dboth components(0000⁰3³)_c,d←(0000⁰2²)_c,d0000⁰2⁰←(0000⁰1¹)_c.By means of the Ritz combination principle the infrared transitions could be used to build up the vibrational energy level scheme of the ν₅ vibrational mode for HCNO and DCNO. The data are only reconcilable with a potential function for ν₅ which exhibits a low barrier opposing linearity. Preliminary values of the potential parameters were obtained using different approximate theoretical approaches.A reinterpretation of the r₈ structure parameters of fulminic acid in the light of the quasilinear model leads to an explanation of the extraordinarily short CH internuclear distance of 1.027 Å as the projection of a CH bond length of 1.060(5) Å upon the heavy-atom axis.The isotopic shift upon deuteration observed in the infrared data indicate that the ν₅ fundamental vibration is primarily an HCN bending motion. The ν₄ fundamental vibration (skeletal bending motion) of HCNO is located at 537 cm^?1 and does not exhibit any hot band structure which would be indicative of a perturbed potential function.     

Targeted determination of the early stage SCLC specific biomarker pro-gastrin-releasing peptide (ProGRP) at clinical concentration levels in human serum using LC-MS

Pro-gastrin-releasing peptide (ProGRP) is used as a specific diagnostic marker for small cell lung cancer (SCLC), a rapidly growing neoplasm with high mortality. Our object was to develop an LC-MS method for the detection and quantification of ProGRP in human serum using the specific tryptic digestion product NLLGLIEAK (m/z 485.8 for [M + 2H](2+)). For this purpose the sample pretreatment, clean-up, enrichment, and LC-MS conditions were evaluated. Sample pretreatment was carried out using ACN precipitation to decrease the sample complexity. Although ProGRP (31-98) standards were soluble in 99% ACN, it showed that optimal signal intensities were obtained by adding ACN to the serum in a 1:1 ratio v/v. A simplified tryptic digest protocol was carried out using 100 mM triethanolamine buffer to ensure pH stability during the whole procedure. The simplified protocol also includes omission of reducing and alkylating reagents. Necessary additional sample clean-up was achieved by trapping NLLGLIEAK on a RAM column (ADS-C8) which was back-flushed onto the analytical BioBasic C8 column. Volume of injection, sample enrichment, and column capacity are among the factors optimized to reach a mass LOD of 150 pg on column (OC) ProGRP (31-98). Detection of ProGRP in the serum sample of a patient suffering from SCLC with a clinically relevant concentration shows the potential of the method in diagnosis, prognosis, and monitoring of the disease.     

Structural studies and T c dependence in La2−x Dy x Ca y Ba2Cu4+y O z type mixed oxide superconductors

A new series of mixed oxide superconductors with the stoichiometric composition La_2−x Dy _x Ca _y Ba₂Cu_4+y O _z (x=0.0 − 0.5, y=2x) has been studied for structural and superconductiong properties. Our earlier studies on La_2−x (Y/Er) _x Ca _y Ba₂Cu_4+y O _z series, show a strong dependence of T _c on hole concentration (p _sh). In the present work, the results of the analysis of the neutron diffraction measurements at room temprerature on x=0.3 and 0.5 samples are reported. It is interesting to know that Ca substitutes for both La and Ba site with concomitant displacement of La onto Ba site. Superconductivity studies show that maximum T _c is obtained for x=0.5, y=1.0 sample (T _c ∼ 75 K), for La_1.5Dy_0.5Ca₁Ba₂Cu₅O _z (La-2125).     

Preconditioning discretizations of systems of partial differential equations

This survey paper is based on three talks given by the second author at the London Mathematical Society Durham Symposium on Computational Linear Algebra for Partial Differential Equations in the summer of 2008. The main focus will be on an abstract approach to the construction of preconditioners for symmetric linear systems in a Hilbert space setting. Typical examples that are covered by this theory are systems of partial differential equations which correspond to saddle point problems. We will argue that the mapping properties of the coefficient operators suggest that block diagonal preconditioners are natural choices for these systems. To illustrate our approach a number of examples will be considered. In particular, parameter‐dependent systems arising in areas like incompressible flow, linear elasticity, and optimal control theory will be studied. The paper contains analysis of several models which have previously been discussed in the literature. However, here each example is discussed with reference to a more unified abstract approach. Copyright © 2010 John Wiley & Sons, Ltd.