Response to Steven Colbert: Spicing Up the Exposition of Differential Equations via Engaging with Relevant History of Algebra

This paper deals with some problems that can be incorporated in the exposition of ordinary differential equations in courses on Differential Equations and System Dynamics or Intermediate Strength of Materials, with a view to promote more interest and excitement by the attendees, both students and lecturers

Differential Equations of Love and Love of Differential Equations

In this paper, simple ordinary differential equations are discussed against the background of William Shakespeare’s Romeo and Juliet. In addition, a version of this relationship in a somewhat opposite setting is considered. It is proposed that engineering mathematics courses include this topic in order to promote additional interest in differential equations. In the final section it is shown that vibration of a single-degree-of-freedom mechanical system can be cast as a love-hate relationship between its displacement and velocity, and dynamic instability identified as a transition from trigonometric love to hyperbolic

Summary of Research Report

Ten papers, published in various publications, on buckling, and the effects of imperfections on various structures are presented. These papers are: (1) Buckling mode localization in elastic plates due to misplacement in the stiffner location; (2) On vibrational imperfection sensitivity on Augusti's model structure in the vicinity of a non-linear static state; (3) Imperfection sensitivity due to elastic moduli in the Roorda Koiter frame; (4) Buckling mode localization in a multi-span periodic structure with a disorder in a single span; (5) Prediction of natural frequency and buckling load variability due to uncertainty in material properties by convex modeling; (6) Derivation of multi-dimensional ellipsoidal convex model for experimental data; (7) Passive control of buckling deformation via Anderson localization phenomenon; (8)Effect of the thickness and initial im perfection on buckling on composite cylindrical shells: asymptotic analysis and numerical results by BOSOR4 and PANDA2; (9) Worst case estimation of homology design by convex analysis; (10) Buckling of structures with uncertain imperfections - Personal perspective

A paradox of non-monotonicity in stability of pipes conveying fluid

The paradoxical result of the non-monotonous relationship between the critical speed of the fluid that is conveyed in the elastic pipe, and the mass ratio was reported first some four decades ago. Since then this result was reproduced in numerous books and articles. In this study the paradox is revisited. It appears that it is a numerical artifact; instead of non-monotonicity there are jumps

On proper applications of Galërkin’s approach in structural mechanics courses

An incautious use of the well-known Galërkin’s technique to find approximate solutions of a differential problem may lead to apparently wrong results. Examples are based on an inverse approach to investigate buckling of compressed axisymmetric circular plates, a common subject in courses on mechanics of structures and stability of structural elements. We discuss how a mistake may originate and show how it is possible to recover the expected results, thus providing a means for the students to cross-check their outputs

Projects-Based Instruction of Intermediate Strength of Materials Course: Preparing Students for Future Workforce

This paper is devoted to the transformative instruction of Intermediate Strength of Materials or Aerospace Structures courses. It is argued that instead of placing heavy emphasis on tests and exams it is preferable to engage students with small size projects covering main topics of the course. Each student is assigned a serial number. The parameters of the loads and/or parameters describing geometric dimensions in offered project problems are made dependent on the serial number. This creates individualized project and takes care that students perform these individually even in case they cooperate. The latter is being welcomed since it promotes discussions between students, thus resulting in the better understanding of the material. Projects create natural interaction between the faculty, teaching assistant, and the students, who pose questions via Canvas system or any other accepted software in use at the given University

Hybrid probabilistic and convex modeling of excitation and response of periodic structures

In this paper, a periodic finite-span beam subjected to the stochastic acoustic pressure with bounded parameters is investigated. Uncertainty parameters exist in this acoustic excitation due to the deviation or imperfection. First, a finite-span beams subjected to the random acoustic pressure field are studied, the exact analytic forms of the cross-spectral density of both the transverse displacement and the bending moment responses of the structure are formulated. The combined probabilistic and convex modeling of acoustic excitation appears to be most suitable, since there is an insufficient information available on the acoustic excitation parameters, to justify the totally probabilitic analysis. Specifically, we postulate that the uncertainty parameters in the acoustic loading belong to a bounded, convex set. In the special case when this convex set is an ellipsoid, closed form solutions are obtained for the most and least favorable mean square responses of both the transverse displacement and bending moment of the structure. Several finite-span beams are exemplified to gain insight into proposal methodology

Sixty years of stochastic linearization technique

Stochastic linearization technique is a versatile method of solving nonlinear stochastic boundary value problems. It allows obtaining estimates of the response of the system when exact solution is unavailable; in contrast to the perturbation technique, its realization does not demand smallness of the parameter; on the other hand, unlike the Monte Carlo simulation it does not involve extensive computational cost. Although its accuracy may be not very high, this is remedied by the fact that the stochastic excitation itself need not be known quite precisely. Although it was advanced about six decades ago, during which several hundreds of papers were written, its foundations, as exposed in many monographs, appear to be still attracting investigators in stochastic dynamics. This study considers the methodological and pedagogical aspects of its exposition

Who needs refined structural theories?

This paper discusses the question posed in the title and available options for the structural analysis of metallic and composite structures concerning the choice of 1D, 2D, and 3D theories. The focus is on the proper modeling of various types of mechanical behaviors and the associated solution’s efficiency. The necessity and convenience of developing higher‐order structural theories are discussed as compared to 3D models. Multiple problems are considered, including linear and nonlinear analyses and static and dynamic settings. Some possible guidelines on the proper selection of a model are outlined, and quantitative estimations on the accuracy are provided. It is demonstrated that the possibility of incorporating higher‐order effects in 1D and 2D models continues to remain attractive in many structural engineering problems to alleviate the computational burdens of 3D analyses

Hybrid theoretical, experimental and numerical study of vibration and buckling of composite shells with scatter in elastic moduli

AbstractHybrid theoretical, experimental and numerical method is proposed for free vibration and buckling of composite shell with unavoidable scatter in elastic moduli. Based on the Goggin’s measurement techniques, the elastic moduli for material T300-QY8911 are measured, and a set of experimental points are obtained. The measurements of elastic moduli are quantified by either (1) the smallest ellipsoid and (2) the smallest four-dimensional uncertainty hyper-rectangle. Then uncertainty propagation in vibration and buckling problems of composite shell by ellipsoidal analysis and interval analysis are, respectively, studied from the theoretical standpoint. Comparison between these analyses is performed numerically