Second-order Perturbation Technique Research Articles

PERTURBATION TECHNIQUE TO APPROXIMATE THE EFFECT OF DAMPING NONPROPORTIONALITY IN MODAL DYNAMIC ANALYSIS

A general second-order perturbation technique is applied in approximating the complex eigenvectors and eigenvalues of MDOF system with moderately nonproportional viscous damping. Only the nonproportionality, not the overall level of damping itself, is assumed to be either moderate or weak. Perturbation coefficients explicitly relate the complex eigenvectors and eigenvalues to the nonproportional damping matrix of the system and the natural frequencies and mode shapes of the counterpart undamped system. Using the perturbed complex “modes”, the dynamic response is expressed in a form analogous to modal superposition for proportionally damped system, but with additional terms explicitly representing nonproportionality effect. Numerical examples are given to illustrate the accuracy of the technique. It is pointed out that once the mode shapes and natural frequencies of the counterpart undamped system are known, in the present technique there is no need for numerically solving another eigenvalue problem, which would be bigger, when the damping is to be considered. This computational advantage is even more significant when designing or else identifying the system damping; either task requires reanalysis everytime that the damping matrix is changed. As a second advantage of the method, additional physical insights into the mathematical analysis are obtained. For example, the mode shapes of the counterpart undamped system are seen to couple to form the complex eigenvectors.

Stochastic Stress Analysis of Assembled Structures

Abstract Intrinsic stress caused by uncertain members in an assembled structure is analyzed by means of the stochastic finite element method. Use is made of the second-order perturbation technique in the formulation of the stochastic finite element method, which is able to deal with uncertainties involved in the stiffness equation. The expectation and variance of the initial stresses in closed ring and portal frame are successfully estimated when the covariance matrix is given for uncertain fluctuation in the dimensions of the curved beam and quasi-straight beam members.

Finite element methods in probabilistic mechanics

Probabilistic methods, synthesizing the power of finite element methods with second-order perturbation techniques, are formulated for linear and nonlinear problems. Random material, geometric properties and loads can be incorporated in these methods, in terms of their fundamental statistics. By construction, these methods are applicable when the scale of randomness is not too large and when the probabilistic density functions have decaying tails. By incorporating certain computational techniques, these methods are shown to be capable of handling large systems with many sources of uncertainties. Applications showing the effects of combined random fields and cyclic loading/stress reversal are studied and compared with Monte Carlo simulation results.

Diffraction losses for symmetrically tilted plane reflectors in open resonators

Diffraction power losses for tilted (open) resonators with both circular and strip flat reflectors are computed for several tilt angles by means of a second-order perturbation technique. Results indicate that the power losses increase as the square of the perturbation parameter. The strips exhibit a greater additional power loss per transit for the TEM(0) mode than for the TEM(1) mode; the opposite is true in the case of the circular disks. Results are applied to the He-Ne 6328-A line.

Theory of two-photon absorption and emission second-order saturation effect

A simplified theory of small-signal two-photon absorption is developed, based on the solution of rate equations involving an infinitely short-living intimediate state. It is shown that in a first approximation this theory leads to results identical to those obtained using second-order perturbation techniques. Moreover, the present approach leads to a built-in saturation effect at high intensities depending on the parameters of the laser and of the absorbing medium. An analysis expression is given for the intensity dependent saturation.

A study of the effects of solvent on overtone frequencies

According to Buckingham's theory on solvent effects the shift caused by solute-solvent interaction should be linearly proportional to the vibrational quantum number of the excited vibrational level which is involved. In order to provide an experimental check on this theory we measured the wave numbers of the OH, NH, or CH fundamentals, their first and second overtones of phenol, 2,6-di-tertiary-butyl-4-methylphenol, N-methylaniline, and chloroform in a series of nonpolar or weakly polar solvents. The results indicate that the agreement with experiment is satisfactory for 2,6-di-tertiary-butyl-4-methylphenol and for phenol in the most weakly associating solvents but not for phenol in more strongly associating solvents, or for N-methylaniline, or chloroform. Buckingham's theory in its original form uses second-order perturbation techniques. We tried to improve the agreement between theory and experiment in taking account of the off-diagonal matrix elements of the quartic potential constant. This way we were able to interpret the main trends observed in the spectra but the need for higher approximations is still indicated. The spectra of 2,6-di-tertiary-butyl-4-methylphenol were also measured at −78°C and, in a solid glass, at −190°C. The observed changes can be interpreted in terms of increased solute-solvent interactions becoming more significant at liquid air temperature.