An improved precise integration method for nonlinear dynamic system

In the present paper, based on the precise integration method established in linear dynamic system, an improved precise integration method is presented for nonlinear dynamic system. Firstly, the nonlinear dynamic system is converted into an augmented Lie type dynamic system. Then the precise integration method is improved for solving the above augmented equation and preserving its group structure in the meantime. Finally, two numerical examples are presented to demonstrate the validity and effectiveness of the proposed method.     

An approximate method of determining the vorticity in the separation region as the viscosity tends to zero

In accordance with the Prandtl—Batchelor theorem, the vorticity in a separation region is constant in a laminar flow with vanishingly small viscosity. Batchelor proposed that the vorticity should be determined by matching the inviscid flow and the boundary layer at the edge of the separation region. An approximate method is constructed and, under a number of simplifying assumptions, used to consider a flow with a separation region in a rectangular trough.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 10–15, January–February, 1982.I thank N. A. Slezkin and the participants of his seminar for several discussions of the work and helpful comments.     

An experimental method for determining the dynamic contact law

An experimental method is investigated whereby the strain response from an impacted beam is sufficient to determine the contacting force. Once the force is known, it is shown how the contact law can be determined. Experimental results for an impacted aluminum beam are demonstrated.     

An analytical method for the optimum thermal design of convective longitudinal fin arrays

The paper analyzes the problem of the optimum thermal design of free and forced convection fin arrays composed of longitudinal fins with constant thickness. Two different optimization problems have been considered: the minimization of the weight for a given heat flow and the maximization of the heat flow for a given weight. Two different geometrical configurations of the fin array have been considered: closed array and open array. The procedure for the optimization is provided in a general case and a complete analytical solution of the problem, for the case with the tip approximated as being insulated, is developed. The paper contains several illustrative examples of the application of the proposed optimization procedure.     

An improved method for the measurement of strong flow birefringence

An improved procedure is proposed to solve the problem of measuring strong birefringence on flowing samples which have a retardation larger than /2 The procedure can be applied to devices which are based on the modulation of the polarization vector with a high-speed rotating half-wave plate. It uses two optical configurations which have different responses to the retardation. By combining the information obtained from the two configurations, one can easily and accurately determine the actual retardation of the sample, irrespective of the quadrant in which the measured retardation is located, provided the sample is isotropic. This technique can also overcome the problem of large errors occurring in the vicinity of = (2m+1)/2 due to the limited sensitivity around these values of . The use of the technique is illustrated with measurements on an isotropic solution of poly(benzyl glutamate) in m-cresol, which is strongly birefringent during shear flow.     

A method for determining the parameters of the hereditary kernels in the nonlinear theory of viscoelasticity

The analytical and experimental procedures of the method for determining the parameters of the hereditary kernels in the modified Rabotnov’s nonlinear viscoelastic model are outlined. The method is based on the similarity of the isochronous creep curves to the instantaneous-deformation curve. The parameters of the kernels are determined by fitting the discrete values of the kernels that are found by differentiating the average similarity function. The discrete values of the kernels in the domain of singularity are calculated using weight functions     

A comparison of two formulations of the fin efficiency for straight fins

A formulation of the fin efficiency based on Newton’s law of cooling is compared with a formulation based on a ratio of heat transferred from the fin surface to the surrounding fluid to the heat conducted through the base.The first formulation requires that the solution of the nonlinear fin equations for constant heat transfer coefficient and constant thermal conductivity is known,whilst the second formulation of the fin efficiency requires only that a first integral of the model equation is known.This paper shows the first formulation of the fin efficiency contains approximation errors as only power series and approximate solutions to the nonlinear fin equations have been determined.The second formulation of the fin efficiency is exact when the first integrals can be determined.     

An approximate method for solving Hopf's equation of Burger's model of turbulence

The Hopf functional equation of Burger's model of turbulence is considered and the Lewis-Kraichnan version of the functional formulation is discussed. An approximate method based on series expansion of natural logarithm of the characteristic functional is presented. The equations governing the hierarchy of cumulants are derived. Several possible methods of closing the system of equations are discussed.Presently academic visitor at the Department of Civil and Geological Engineering, Princeton University, Princeton, NJ, USA.     

An approximate solution of some spatial physical nonlinear problems of the theory of elasticity

International Applied Mechanics -