Two systems for automatic reduction of time-dependent photomechanics data

Time-dependent photomechanical model materials will not, in general, exhibit a one-to-one relationship between the instantaneous stress and the relative retardation. Due to the complicated nature of this relationship, it is often more accurate to use the analog method of data reduction. This method consists of reproducing the fringe-order history observed in the model in a tensile specimen of the model material and recording the corresponding stress history. Two servo systems are described in this report which may be programmed to produce a desired fringe-order history in a time-dependent tensile specimen.     

Two photoviscoelasticity experiments using analog data reduction

A tensile analog may conveniently be used to determine the history of the principal-stress difference from the results of some photoviscoelastic model tests. With this method, the model test is run, and the fringe-order history at the point (s) of interest in the model is recorded. A tensile specimen of the same material is placed in a loading frame and loaded with a stress history which forces it through the fringe-order history observed in the model. This stress history is then the desired principal-stress-difference history occurring in the model. The stress history at points in two nonhomogeneous models was determined with a tensile analog. The uses and limitations of the analog technique are discussed.     

A photomechanics material for elastoplastic stress analysis

The object of this investigation was to develop a technique or method for elastoplastic stress analysis using the optical effects of transparent materials. Of paramount importance was the selection and characterization of a suitable model material. In particular, it was desirable that the material be able to undergo large plastic strains while, at the same time, exhibiting a suitable level of optical response. A mixture of flexible and rigid polyester resins was found suitable, i.e., the mixture exhibited large strains and good optical response. It was found that unload birefringence (fringe order immediately upon removal of load) could be used to determine strain for a uniaxial-stress field. In particular, it provided a means for evaluating stress- and strain-concentration factors. Comparisons with other methods showed that the proposed method was reliable and gave results that are similar to those by other means. The usefulness of the material and method for two- and three-dimensional problems awaits further study.     

A simple holographic interferometer for static and dynamic photomechanics

A particular variation of holographic imaging system is described which, when used as a multiple-exposure holographic interferometer, possesses advantages for applications in static and dynamic photomechanics. Large fields of view can be obtained. Rigid-body motions produced by loading are automatically eliminated. The holograms can be recorded on medium-resolution films which have high sensitivity. Specimens manufactured with readily available materials can be used for the determination of isochromatics and isopachics. An attempt is made to describe this contribution in the background of previous developments in interferometry. Operation of the system is interpreted by showing the equivalence of the holographic interferometer to a combination of two systems presently in use in experimental stress analysis: a Fizeau interferometer and an optical spatial filter. The interpretation of isochromatics and isopachis as moiré phenomena is emphasized. Isochromatics and isopachics are presented as illustrations of the applications of the method to the solution of static-stress problems, and they are used in the solution of some not yet solved dynamic-stress problems. Whole-field static isochromatics obtained as absolute-retardation interference are shown. Also shown are whole-field dynamic isopachics.     

Uncertainty quantification for stochastic dynamical systems using time-dependent stochastic bases

A novel method based on time-dependent stochastic orthogonal bases for stochastic response surface approximation is proposed to overcome the problem of significant errors in the utilization of the generalized polynomial chaos(GPC) method that approximates the stochastic response by orthogonal polynomials. The accuracy and effectiveness of the method are illustrated by different numerical examples including both linear and nonlinear problems. The results indicate that the proposed method modifies the stochastic bases adaptively, and has a better approximation for the probability density function in contrast to the GPC method.     

Measurement of elastic-plastic stresses by scattered-light photomechanics

Several birefringent materials were studied for their suitability for use in three-dimensional photoplasticity. This study resulted in the selection of cellulose propionate as model material. Its close match in index of refraction with ordinary mineral oil makes cellulose propionate suitable for scattered-light photomechanics. Viscoelastic behavior of the material is used to simulate elastic-plastic behavior of metals. The stress, strain and optical behavior of the material has been studied under slow loading in finite steps. A successful solution of an elastic-plastic torsion problem was obtained, demonstrating the applicability of the techniques of scattered-light photoplasticity to three-dimensional problems. This experimental method does not require the unloading and slicing of the model, avoiding any errors that might be introduced by residual stresses due to unloading. A major advantage of the method is the use of live loading, which allows the investigation of several load levels with the same model. Stress- and strain-concentration factors for the grooved shaft in torsion showed excellent agreement with Neuber's analytical results. Distributions of shear stress and shear strain across the minimum section of the model were compared to elastic theory. Integration of the shear-stress distributions showed good agreement with the measured values of applied torque.     

Theory of the absolute stability of time-dependent systems

S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 31, No. 10, pp. 77–80, October, 1995.     

Size reduction methods for the implicit time-dependent simulation of micro–macro viscoelastic flow problems

Traditionally, micro–macro simulations have been performed using simple explicit time-marching algorithms, which lack the desirable stability of implicit methods. In this study, a fully implicit time integration scheme is presented and implemented for the first time for the solution of time-dependent complex flows using the Brownian Configuration Fields approach. Special techniques need to be applied to deal with the very large size of the resulting linear systems. A novel size-reduction scheme is used, allowing an independent treatment for each molecular field and suited to parallel hardware architecture. To illustrate the method, a selected number of applications using linear spring chains are presented and the results are compared with their corresponding closed form constitutive equation. The excellent agreement between the results demonstrates the feasibility of the proposed approach.     

Conservation laws and reduction to quadratures of the generalized time-dependent duffing equation

The field-momentum method and the method of Hamilton and Jacobi are applied for finding the conservation laws and reduction to quadratures of the non-linear time-dependent equation

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where Q_i are constant parameters and f_i(t) are supposed to be power-type functions of time, that is, f_i(t) = t^k_i.     

Finite element procedures for time-dependent convection–diffusion–reaction systems

New finite element procedures based on the streamline-upwind/Petrov-Galerkin formulations are developed for time-dependent convection-diffusion-reaction equations. These procedures minimize spurious oscillations for convection-dominated and reaction-dominated problems. The results obtained for representative numerical examples are accurate with minimal oscillations. As a special application problem, the single-well chemical tracer test (a procedure for measuring oil remaining in a depleted field) is simulated numerically. The results show the importance of temperature effects on the interpreted value of residual oil saturation from such tests.     

Fast data-driven model reduction for nonlinear dynamical systems

We present a fast method for nonlinear data-driven model reduction of dynamical systems onto their slowest nonresonant spectral submanifolds (SSMs). While the recently proposed reduced-order modeling method SSMLearn uses implicit optimization to fit a spectral submanifold to data and reduce the dynamics to a normal form, here, we reformulate these tasks as explicit problems under certain simplifying assumptions. In addition, we provide a novel method for timelag selection when delay-embedding signals from multimodal systems. We show that our alternative approach to data-driven SSM construction yields accurate and sparse rigorous models for essentially nonlinear (or non-linearizable) dynamics on both numerical and experimental datasets. Aside from a major reduction in complexity, our new method allows an increase in the training data dimensionality by several orders of magnitude. This promises to extend data-driven, SSM-based modeling to problems with hundreds of thousands of degrees of freedom.

     

Control of linear instabilities by dynamically consistent order reduction on optimally time-dependent modes

Nonlinear Dynamics - Identification and control of transient instabilities in high-dimensional dynamical systems remain a challenge because transient (non-normal) growth cannot be accurately...     

A new model reduction method for nonlinear dynamical systems

The present research work proposes a new systematic approach to the problem of model-reduction for nonlinear dynamical systems. The formulation of the problem is conveniently realized through a system of singular first-order quasi-linear invariance partial differential equations (PDEs), and a rather general explicit set of conditions for solvability is derived. In particular, within the class of analytic solutions, the aforementioned set of conditions guarantees the existence and uniqueness of a locally analytic solution. The solution to the above system of singular PDEs is then proven to represent the slow invariant manifold of the nonlinear dynamical system under consideration exponentially attracting all dynamic trajectories. As a result, an exact reduced-order model for the nonlinear system dynamics is obtained through the restriction of the original system dynamics on the aforementioned slow manifold. The local analyticity property of the solution’s graph that corresponds to the system’s slow manifold enables the development of a series solution method, which allows the polynomial approximation of the system dynamics on the slow manifold up to the desired degree of accuracy and can be easily implemented with the aid of a symbolic software package such as MAPLE. Finally, the proposed approach and method is evaluated through an illustrative biological reactor example.     

Model reduction for constrained mechanical systems via spectral submanifolds

Nonlinear Dynamics - Dynamical systems are often subject to algebraic constraints in conjunction with their governing ordinary differential equations. In particular, multibody systems are commonly...     

A multiple-pulse ruby-laser system for dynamic photomechanics: Applications to transmitted- and scattered-light photoelasticity

An ultrahigh-speed multiple-frame recording system for two- and three-dimensional dynamic photomechanics has been developed and is described here. The output from a ruby laser is modulated with a Pockels cell to produce a train of short, intense, monochromatic and polarized light pulses. Pulse widths of 50 nsec and repetition rates of up to 170,000 pulses/sec are obtained. These light pulses are synchronized with a “smear camera” and the event to produce a multiple-frame record of the phenomenon. The simplified camera requirements necessary for this purpose are indicated. The system is demonstrated by recording two-dimensional dynamic and scattered-light isochromatic fringe patterns. The capability of multiple recording of scattered-light fringe patterns, achieved here for the first time, has a tremendous potential for three-dimensional dynamic stress analysis. The developed system is also well suited for dynamic moiré, interferometry and holography.     

Nonlinear integral resonant controller for vibration reduction in nonlinear systems

A new nonlinear integral resonant controller (NIRC) is introduced in this paper to suppress vibration in nonlinear oscillatory smart structures. The NIRC consists of a first-order resonant integrator that provides additional damping in a closed-loop system response to reduce high-amplitude nonlinear vibration around the fundamental reso-nance frequency. The method of multiple scales is used to obtain an approximate solution for the closed-loop system. Then closed-loop system stability is investigated using the resulting modulation equation. Finally, the effects of different control system parameters are illustrated and an approximate solution response is verified via numerical simulation results. The advantages and disadvantages of the proposed controller are presented and extensively discussed in the results. The controlled system via the NIRC shows no high-amplitude peaks in the neighboring frequencies of the resonant mode, unlike conventional second-order compensation methods. This makes the NIRC controlled system robust to excitation frequency variations.