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

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

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

An interesting material that appears to be fit to possibly all future mechanical vibration textbooks

A material is suggested for future mechanical vibration textbooks. Both mathematically and conceptually it is simpler than most of the material that is already included in the existing textbooks. It pertains to the inverse vibration problem for inhomogeneous beam, i.e. the beam with the modulus of elasticity that varies along the axial coordinate. Specifically, the solution of the following problem is presented: Find a distribution of the modulus of elasticity of an inhomogeneous beam such that the beam would possess the preselected simple, polynomial vibration mode shape

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

dynamic analysis of bernoulli euler beams with interval uncertainties under moving loads

Abstract This paper deals with the dynamic analysis of Bernoulli-Euler beams with uncertain parameters crossed by moving loads. Uncertainties associated with mass density, Young's modulus and load velocity are modeled as interval variables with given lower bounds and upper bounds. In order to evaluate the bounds of the interval displacement of the beam, an efficient procedure based on the use of the classical modal superposition method in conjunction with the improved interval analysis via extra unitary interval is developed. The key idea is to seek the bounds of the response by exploring just a few combinations of the endpoints of the uncertain parameters appropriately selected based on those yielding the minimum and maximum values of the interval natural frequencies

Applicability and Limitations of Simplified Elastic Shell Theories for Vibration Modelling of Double-Walled Carbon Nanotubes

The applicability and limitations of simplifiedmodels of thin elastic circular cylindrical shells for linear vibrations of double-walled carbon nanotubes (DWCNTs) are considered. The simplified models, which are based on the assumptions of membrane and moment approximate thin-shell theories, are compared with the extended Sanders–Koiter shell theory. Actual discrete DWCNTs are modelled by means of couples of concentric equivalent continuous thin, circular cylindrical shells. Van der Waals interaction forces between the layers are taken into account by adopting He’s model. Simply supported and free–free boundary conditions are applied. The Rayleigh–Ritz method is considered to obtain approximate natural frequencies and mode shapes. Different aspect and thickness ratios, and numbers of waves along longitudinal and circumferential directions, are analysed. In the cases of axisymmetric and beam-like modes, it is proven that membrane shell theory, differently from moment shell theory, provides results with excellent agreement with the extended Sanders–Koiter shell theory. On the other hand, in the case of shell-like modes, it is found that both membrane and moment shell theories provide results reporting acceptable agreement with the extended Sanders–Koiter shell theory only for very limited ranges of geometries and wavenumbers. Conversely, for shell-like modes it is found that a newly developed, simplified shell model, based on the combination of membrane and semi-moment theories, provides results in satisfactory agreement with the extended Sanders–Koiter shell theory in all ranges