Reinvestigation on Assessing the Stability of Mullagulov Tested Steel Rods under Follower Forces

Dynamic instability is an interesting topic in the mechanics of elastic structures. Though the subject has been formed by many analytical, numerical, and experimental investigations, it has many issues, as evidenced from the critical overview of Elishakoff. Furthermore, the controversial articles of Koiter and Sugiyama on unrealistic and realistic follower forces demand experimental verification. Mullagulov has proposed a device for creating the follower forces and tested steel rods under compression. This paper highlights the experimentation of Mullagulov and his observations briefly to examine the influence of material properties on the stability load estimations and to confirm the practical realization of follower forces.

Corotational formulation for dynamic analysis of flexible offshore structures

Ο στόχος αυτής της έρευνας είναι να αναπτυχθεί ένα ακριβές και αποτελεσματικό συμπεριστροφικό στοιχείο δοκού για τη δυναμική ανάλυση εύκαμπτων ϑαλασσίων κατασκευών. Η στατική διατύπωση της συμπεριστροφικής δοκού βασίζεται κυρίως στο μοντέλο του [60], εφαρμόζοντας ένα διαφορετικό σύστημα συντεταγμένων που οδηγεί σε διαφορετικές εκφράσεις της ελαστικής δύναμης και του μητρώου δυσκαμψίας. Ο τύπος του στοιχείου που εφαρμόζεται είναι ο Bernoulli. Συνεισφορά της παρούσας διατριβής όσον αφορά τη στατική συμπεριστροφική δοκό είναι ότι μελετώνται ακόλουθες δυνάμεις (follower forces) και για τον λόγο αυτό καταστρώνεται το σχετικό μητρώο διόρθωσης. Η δυνατότητα εφαρμογής της προαναφερθείσας ανάπτυξης αποδεικνύεται μέσω ενός αριθμητικού προβλήματος όπου επιτυγχάνεται καλή σύγκριση με τα μοντέλα άλλων δύο ερευνητών. Καταστρώνεται ένα γενικό και αποτελεσματικό δυναμικό μοντέλο για μια χωρική εύκαμπτη δοκό το οποίο είναι ανεξάρτητο από τις ϑεωρίες που χρησιμοποιούνται για την περιγραφή των μεγάλων μετατοπίσεων και των παραμορφώσεων. Γενικά, στις ϑεωρίες μεγάλων μετατοπίσεων εμπλέκεται ένα σύστημα συντεταγμένων που οδηγεί στη συμπερίληψη ενός Coriolis/γυροσκοπικού πίνακα στην έκφραση της δύναμης αδράνειας του σώματος. Μια συγκεκριμένη μορφή αυτού του πίνακα χρησιμοποιείται για την παραγωγή συνοπτικών εκφράσεων για τους δυναμικούς όρους (δύναμη αδράνειας και δυναμικό εφαπτομενικό μητρώο) που οδηγεί σε αποδοτικό αναλυτικό αλλά και αριθμητικό μοντέλο. Σε επόμενο βήμα, το γενικό δυναμικό μοντέλο εφαρμόζεται στη συμπεριστροφική δοκό και όλοι οι δυναμικοί όροι υπολογίζονται ειδικά για το στοιχείο αυτό. Για την συμπεριστροφική δοκό επιτυγχάνεται περαιτέρω βελτίωση της αποτελεσματικότητας του μοντέλου. Συγκεκριμένα, όλα τα ολοκληρώματα υπολογίζονται αναλυτικά και, επομένως, δεν απαιτείται αριθμητική ολοκλήρωση. Τέλος, στους δυναμικούς πίνακες, οι όροι που αποδεικνύεται ότι δεν είναι σημαντικοί αφαιρούνται από τη διατύπωση. Το προτεινόμενο μοντέλο δίνει καλή σύγκριση με το συμπεριστροφικό μοντέλο του [50] και αποδεικνύεται πιο αποτελεσματικό, καθώς απαιτεί λιγότερο υπολογιστικό χρόνο για διάφορα δυναμικά παραδείγματα. Τέλος, το προτεινόμενο μοντέλο της δυναμικής συμπεριστροφικής δοκού επεκτείνεται για την ανάλυση της κίνησης της δοκού εντός κινούμενου ρευστού εισάγοντας μια τροποποιημένη εξίσωση Morison που υπολογίζει την υδροδυναμική δύναμη σε μια δοκό οποιουδήποτε προσανατολισμού. Η αναφορά [81] αφορά σε ένα υδροδυναμικό μοντέλο όπου παρουσιάζονται οι υδροδυναμικοί όροι για το στοιχείο συμπεριστροφικής δοκού του [26] το οποίο όμως μπορεί να είναι μόνο βυθισμένο στο νερό. Στην παρούσα διατριβή, η κατάστρωση των δυναμικών όρων αφορά τόσο τα βυθισμένα στοιχεία όσο και τα στοιχεία που διαπερνούν την ελεύθερη επιφάνεια του νερού. Για την επικύρωση του παρόντος συμπεριστροφικού και ρευστοδυναμικού μοντέλου, πραγματοποιήθηκαν πειράματα μικρής κλίμακας σχετικά με μια δοκό που υπόκειται σε μεγάλες περιστροφές υπό τη φόρτιση ανέμου. Τα προαναφερθέντα πειράματα καθώς και άλλα προβλήματα αλληλεπίδρασης ρευστού/στερεού χρησιμοποιήθηκαν για την εξέταση της αποτελεσματικότητας του προτεινόμενου μοντέλου. Τα αριθμητικά αποτελέσματα αυτής της έρευνας συγκρίνονται με τα πειραματικά αποτελέσματα καθώς και με αυτά του λογισμικού OrcaFlex και η σύγκριση είναι αρκετά καλή.

Flapping, swirling and flipping: Non-linear dynamics of pre-stressed active filaments

Initially straight slender elastic filaments and rods with geometrically constrained ends buckle and form stable two-dimensional shapes when compressed by bringing the ends together. It is known that beyond a critical value of this pre-stress, clamped rods transition to bent, twisted three-dimensional equilibrium shapes. Here, we analyze the three-dimensional instabilities and dynamics of such pre-stressed, initially twisted filaments subject to active follower forces and dissipative fluid drag. We find that degree of boundary constraint and the directionality of active forces determines if oscillatory instabilities can arise. When filaments are clamped at one end and pinned at the other with follower forces directed towards the clamped end, stable planar flapping oscillations result; reversing the directionality of the active forces quenches the instability. When both ends are clamped however, computations reveal a novel secondary instability wherein planar oscillations are destabilized by off-planar perturbations resulting in three-dimensional swirling patterns with periodic flips. These swirl-flip transitions are characterized by two distinct and time-scales. The first corresponds to unidirectional swirling rotation around the end-to-end axis. The second captures the time between flipping events when the direction of swirling reverses. We find that this spatiotemporal dance resembles relaxation oscillations with each cycle initiated by a sudden jump in torsional deformation and then followed by a period of gradual decrease in net torsion until the next cycle of variations. Our work reveals the rich tapestry of spatiotemporal patterns when weakly inertial strongly damped rods are deformed by non-conservative active forces. Practically, our results suggest avenues by which pre-stress, elasticity and activity may be used to design synthetic fluidic elements to pump or mix fluid at macroscopic length scales.

Buckling instabilities and spatio-temporal dynamics of active elastic filaments

Biological filaments driven by molecular motors tend to experience tangential propulsive forces also known as active follower forces. When such a filament encounters an obstacle, it deforms, which reorients its follower forces and alters its entire motion. If the filament pushes a cargo, the friction on the cargo can be enough to deform the filament, thus affecting the transport properties of the cargo. Motivated by cytoskeletal filament motility assays, we study the dynamic buckling instabilities of a two-dimensional slender elastic filament driven through a dissipative medium by tangential propulsive forces in the presence of obstacles or cargo. We observe two distinct instabilities. When the filament’s head is pinned or experiences significant translational but little rotational drag from its cargo, it buckles into a steadily rotating coiled state. When it is clamped or experiences both significant translational and rotational drag from its cargo, it buckles into a periodically beating, overall translating state. Using minimal analytically tractable models, linear stability theory and fully nonlinear computations, we study the onset of each buckling instability, characterize each buckled state, and map out the phase diagram of the system. Finally, we use particle-based Brownian dynamics simulations to show our main results are robust to moderate noise and steric repulsion. Overall, our results provide a unified framework to understand the dynamics of tangentially propelled filaments and filament-cargo assemblies.

Elastohydrodynamical instabilities of active filaments, arrays and carpets analyzed using slender body theory

ABSTRACTThe rhythmic motions and wave-like planar oscillations in filamentous soft structures are ubiquitous in biology. Inspired by these, recent work has focused on the creation of synthetic colloid-based active mimics that can be used to move, transport cargo, and generate fluid flows. Underlying the functionality of these mimics is the coupling between elasticity, geometry, dissipation due to the fluid, and active force or moment generated by the system. Here, we use slender body theory to analyze the linear stability of a subset of these - active elastic filaments, filament arrays and filament carpets - animated by follower forces. Follower forces can be external or internal forces that always act along the filament contour. The application of slender body theory enables the accurate inclusion of hydrodynamic effects, screening due to boundaries, and interactions between filaments. We first study the stability of fixed and freely suspended sphere-filament assemblies, calculate neutral stability curves separating stable oscillatory states from stable straight states, and quantify the frequency of emergent oscillations. When shadowing effects due to the physical presence of the spherical boundary are taken into account, the results from the slender body theory differ from that obtained using local resistivity theory. Next, we examine the onset of instabilities in a small cluster of filaments attached to a wall and examine how the critical force for onset of instability and the frequency of sustained oscillations depend on the number of filaments and the spacing between the filaments. Our results emphasize the role of hydrodynamic interactions in driving the system towards perfectly in-phase or perfectly out of phase responses depending on the nature of the instability. Specifically, the first bifurcation corresponds to filaments oscillating in-phase with each other. We then extend our analysis to filamentous (line) array and (square) carpets of filaments and investigate the variation of the critical parameters for the onset of oscillations and the frequency of oscillations on the inter-filament spacing. The square carpet also produces a uniform flow at infinity and we determine the ratio of the mean-squared flow at infinity to the energy input by active forces. We conclude by analyzing the bending and buckling instabilities of a straight passive filament attached to a wall and placed in a viscous stagnant flow - a problem related to the growth of biofilms, and also to mechanosensing in passive cilia and microvilli. Taken together, our results provide the foundation for more detailed non-linear analyses of spatiotemporal patterns in active filament systems.

Transverse Vibrations of Annular Spinning Disk Subjected to Out-of-Plane Loads and Uniform In-Plane Follower Edge Forces

This paper investigated the transverse vibration natural frequency and stability of an annular spinning disk subjected to concentrated uniform in-plane edge follower forces and out-of-plane loads including mass, damper, and spring. Linear plate theory is used to derive the plate model used in this research, both symmetric in-plane stress fields, due to plate rotation, and asymmetric in-plane stress fields, due to uniform edge follower forces, are considered. The asymmetric stress, which is a function of the rotation of the plate, is obtained from the steady-state response of the coupled in-plane vibration equations. The concentrated in-plane follower edge forces are expanded as Fourier series and the produced stress fields are also achieved from the corresponding Fourier components. To explore the system stabilities, natural frequency variations, in-plane and out-of-plane loads coupling effects, and the instability types, the eigenvalues are plotted with respect to rotation speeds. Based on the eigenvalue analysis, the effects of concentrated edge uniform follower forces on a spinning disk system with out-of-plane loads were concluded.

Instabilities and Spatiotemporal Dynamics of Active Elastic Filaments

Biological filaments driven by molecular motors tend to experience tangential propulsive forces also known as active follower forces. When such a filament encounters an obstacle, it deforms, which reorients its follower forces and alters its entire motion. If the filament pushes a cargo, the friction on the cargo can be enough to deform the filament, thus affecting the transport properties of the cargo. Motivated by cytoskeletal filament motility assays, we study the dynamic buckling instabilities of a two-dimensional slender elastic filament driven through a dissipative medium by tangential propulsive forces in the presence of obstacles or cargo. We observe two distinct instabilities. When the filament’s head is pinned or experiences significant translational but little rotational drag from its cargo, it buckles into a steadily rotating coiled state. When it is clamped or experiences both significant translational and rotational drag from its cargo, it buckles into a periodically beating, overall translating state. Using minimal analytically tractable models, linear stability theory, and fully non-linear computations, we study the onset of each buckling instability, characterize each buckled state, and map out the phase diagram of the system. Finally, we use particle-based Brownian dynamics simulations to show our main results are robust to moderate noise and steric repulsion. Overall, our results provide a unified framework to understand the dynamics of tangentially propelled filaments and filament-cargo assemblies.