Three Geometrically Exact Beam Theories for Analysis of Highly Flexible 1-D Structures

2011 ◽  
Author(s):  
P Frank Pai
Keyword(s):  
Geometrically Exact Beam ◽  
Beam Theories

Dynamic Analysis of a Rotating Double-Tapered FGM Beam with Various Shear Models Using a Constrained Variational Method

2021 ◽  
Author(s):  
Zixuan Zhou ◽  
Xiuchang Huang ◽  
Hongxing Hua
Keyword(s):  
Variational Method ◽  
Shear Deformation ◽  
Power Law ◽  
Slenderness Ratio ◽  
Mathematical Formulation ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Functionally Graded ◽  
Geometrically Exact Beam ◽  
Beam Theories

A constrained variation modeling method for free vibration analysis of rotating double tapered functionally graded beams with different shear deformation beam theories is proposed in this paper. The material properties of the beam are supposed to continuously vary in the width direction with power-law exponent for different indexes. The mathematical formulation is developed based on the geometrically exact beam theory for each beam segment, the admissible functions denoting motion quantities are then expressed by a series of Chebyshev orthogonal polynomials. The governing equations are eventually derived using the constrained variational method to involve the continuity conditions of adjacent segments. Different shear deformation beam theories have been incorporated in the formulations, and the nonlinear effect of bending–stretching coupling vibration together with the Coriolis effect is taken into account. Comparison of dimensionless natural frequencies is performed with the existing literature to ensure the accuracy and reliability of the proposed method. Comparative discussions are performed on the vibration behaviors of the double tapered rotating functionally graded beam with first-order shear deformation beam theory and other higher-order shear deformation beam theories. The effect of material property graduation, power-law index, rotation speed, hub radius, slenderness ratio, and taper ratios is scrutinized via parametric studies, respectively.

scholarly journals Aeroelastic Stability Analysis of Electric Aircraft Wings with Distributed Electric Propulsors

Aerospace ◽  
2021 ◽  
Vol 8 (4) ◽  
pp. 100
Author(s):  
Mohammadreza Amoozgar ◽  
Michael I. Friswell ◽  
Seyed Ahmad Fazelzadeh ◽  
Hamed Haddad Khodaparast ◽  
Abbas Mazidi ◽  
...  
Keyword(s):  
Angular Momentum ◽  
Electric Propulsion ◽  
Aeroelastic Stability ◽  
Concentrated Mass ◽  
High Lift ◽  
Aircraft Wings ◽  
Time Space ◽  
Beam Equations ◽  
Geometrically Exact Beam ◽  
Electric Aircraft

In this paper, the effect of distributed electric propulsion on the aeroelastic stability of an electric aircraft wing was investigated. All the electric propulsors, which are of different properties, are attached to the wing of the aircraft in different positions. The wing structural dynamics was modelled by using geometrically exact beam equations, while the aerodynamic loads were simulated by using an unsteady aerodynamic theory. The electric propulsors were modelled by using a concentrated mass attached to the wing, and the motor’s thrust and angular momentum were taken into account. The thrust of each propulsor was modelled as a follower force acting exactly at the centre of gravity of the propulsor. The nonlinear aeroelastic governing equations were discretised using a time–space scheme, and the obtained results were verified against available results and very good agreement was observed. Two case studies were considered throughout the paper, resembling two flight conditions of the electric aircraft. The numerical results show that the tip propulsor thrust, mass, and angular momentum had the most impact on the aeroelastic stability of the wing. In addition, it was observed that the high-lift motors had a minimal effect on the aeroelastic stability of the wing.

scholarly journals A Numerical Solution for Modelling Mooring Dynamics, Including Bending and Shearing Effects, Using a Geometrically Exact Beam Model

2021 ◽  
Vol 9 (5) ◽  
pp. 486
Author(s):  
Tobias Martin ◽  
Hans Bihs
Keyword(s):  
Numerical Solution ◽  
Offshore Wind ◽  
Mooring Line ◽  
Beam Model ◽  
Tight Coupling ◽  
Energy Propagation ◽  
Two Phase ◽  
Geometrically Exact Beam ◽  
Mooring Dynamics ◽  
Rotational Deformation

During the operation of moored, floating devices in the renewable energy sector, the tight coupling between the mooring system and floater motion results in snap load conditions. Before snap events occur, the mooring line is typically slack. Here, the mechanism of energy propagation changes from axial to bending dominant, and the correct modelling of the rotational deformation of the lines becomes important. In this paper, a new numerical solution for modelling the mooring dynamics that includes bending and shearing effects is proposed for this purpose. The approach is based on a geometrically exact beam model and quaternion representations for the rotational deformations. Further, the model is coupled to a two-phase numerical wave tank to simulate the motion of a moored, floating offshore wind platform in waves. A good agreement between the proposed numerical model and reference solutions was found. The influence of the bending stiffness on the motion of the structure was studied subsequently. We found that increased stiffness increased the amplitudes of the heave and surge motion, whereas the motion frequencies were less altered.

scholarly journals A Refined Theory for Bending Vibratory Analysis of Thick Functionally Graded Beams

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1422
Author(s):  
Youssef Boutahar ◽  
Nadhir Lebaal ◽  
David Bassir
Keyword(s):  
Beam Theory ◽  
Functionally Graded ◽  
Effective Characteristics ◽  
Hyperbolic Function ◽  
Transverse Displacement ◽  
Refined Theory ◽  
Distribution Law ◽  
Fg Beam ◽  
The Impact ◽  
Beam Theories

A refined beam theory that takes the thickness-stretching into account is presented in this study for the bending vibratory behavior analysis of thick functionally graded (FG) beams. In this theory, the number of unknowns is reduced to four instead of five in the other approaches. Transverse displacement is expressed through a hyperbolic function and subdivided into bending, shear, and thickness-stretching components. The number of unknowns is reduced, which involves a decrease in the number of the governing equation. The boundary conditions at the top and bottom FG beam faces are satisfied without any shear correction factor. According to a distribution law, effective characteristics of FG beam material change continuously in the thickness direction depending on the constituent’s volume proportion. Equations of motion are obtained from Hamilton’s principle and are solved by assuming the Navier’s solution type, for the case of a supported FG beam that is transversely loaded. The numerical results obtained are exposed and analyzed in detail to verify the validity of the current theory and prove the influence of the material composition, geometry, and shear deformation on the vibratory responses of FG beams, showing the impact of normal deformation on these responses which is neglected in most of the beam theories. The obtained results are compared with those predicted by other beam theories. It can be concluded that the present theory is not only accurate but also simple in predicting the bending and free vibration responses of FG beams.

AI-Timoshenko: Automatedly Discovering Simplified Governing Equations for Applied Mechanics Problems from Simulated Data

2021 ◽  
pp. 1-37
Author(s):  
Zhanchao Huang ◽  
Chunjiang Li ◽  
Z. L. Huang ◽  
Yong Wang ◽  
Hanqing Jiang
Keyword(s):  
Finite Element Method ◽  
Simulated Data ◽  
Data Driven ◽  
Trial And Error ◽  
Applied Mechanics ◽  
Governing Equations ◽  
Displacement Fields ◽  
3D Finite Element ◽  
3D Finite Element Method ◽  
Beam Theories

Abstract The simplified governing equations of applied mechanics play a pivotal role and were derived based on ingenious assumptions or hypotheses regarding the displacement fields for specific problems. In this paper, we introduce a data-driven method by the name AI-Timoshenko in honor of Timoshenko, father of applied mechanics, to automatically discover simplified governing equations for applied mechanics problems directly from discrete data simulated by the 3D finite element method. This liberates applied mechanicians from burdensome labor, including assumptions, derivation, and trial and error. The simplified governing equations for Euler-Bernoulli and Timoshenko beam theories are successfully rediscovered using the present AI-Timoshenko method, which shows that this method is capable of discovering simplified governing equations for applied mechanics problems.

Linear and non-linear elastic stability of beams by higher order beam theories

2019 ◽  
Author(s):  
Αμαλία Αργυρίδη
Keyword(s):  
Elastic Stability ◽  
Higher Order ◽  
Linear Elastic ◽  
Non Linear ◽  
Fortran 90 ◽  
Beam Theories

Στην παρούσα διδακτορική διατριβή διερευνάται η γραμμική και μη γραμμική ελαστική ευστάθεια ράβδων διαμέσου θεωριών δοκού ανώτερης τάξης. Προκειμένου να υλοποιηθεί αυτό, το πρώτο βήμα της παρούσας διδακτορικής διατριβής είναι η εξέταση του στρεπτοκαμπτικού λυγισμού σύμμικτων δοκών λαμβάνοντας υπόψη τα φαινόμενα της στρέβλωσης (εκτός επιπέδου παραμόρφωση) και την διατμητικής υστέρησης (διαφοροποίηση της κλασσικής κατανομής ορθών τάσεων) λόγω κάμψης και στρέψης. Στο δεύτερο βήμα της παρούσας διδακτορικής διατριβής, μορφώνεται το πρόβλημα της γραμμικής κα μη γραμμικής στατικής ανάλυσης ομογενών δοκών λαμβάνοντας υπόψη τα φαινόμενα της αξονικής στρέβλωσης και της διαστρέβλωσης (εντός επιπέδου παραμόρφωση) επιπρόσθετα σε εκείνα λόγω της καμπτικής και στρεπτικής συμπεριφοράς της δοκού διαμέσου του σχήματος διαδοχικής ισορροπίας που υιοθετείται (πλεονεκτήματα έναντι προβλημάτων ιδιοτιμών). Κύριο τμήμα του δεύτερου βήματος αποτελεί η διατύπωση, η αρχικοποίηση και η επίλυση των προβλημάτων συνοριακών τιμών που αφορούν στον υπολογισμό των αξονικών μορφών στρέβλωσης και διαστρέβλωσης διαμέσου του σχήματος διαδοχικής ισορροπίας. Στο προτελευταίο βήμα της παρούσας διδακτορικής διατριβής, οι προηγούμενα αναπτυγμένες αξονικές μορφές στρέβλωσης και διαστρέβλωσης μαζί με τις αντίστοιχες καμπτικές και στρεπτικές χρησιμοποιούνται προκειμένου να μορφωθεί και να επιλυθεί το πρόβλημα της γραμμικής ελαστικής ευστάθειας δοκών. Στο τελευταίο βήμα της παρούσας διδακτορικής διατριβής, διεξάγεται μη γραμμική ανάλυση λαμβάνοντας υπόψη τα φαινόμενα της στρέβλωσης και της διαστρέβλωσης λόγω αξονικής, διατμητικής, καμπτικής και στρεπτικής δομικής συμπεριφοράς και πραγματοποιείται σύγκριση με τα αντίστοιχα αποτελέσματα που λαμβάνονται στην περίπτωση όπου λαμβάνεται υπόψη μόνο η στρέβλωση και η διατμητική υστέρηση λόγω κάμψης και στρέψης. Αξίζει να σημειωθεί ότι σε όλες τις παραπάνω περιπτώσεις η διατομή της δοκού είναι τυχούσα, ενώ ο λόγος του Poisson λαμβάνεται υπόψη στον υπολογισμό των αξονικών, καμπτικών και στρεπτικών μορφών στρέβλωσης και διαστρέβλωσης. Στην παρούσα διδακτορική διατριβή χρησιμοποιούνται δύο αριθμητικές μέθοδοι. Η πρώτη είναι η μέθοδος των συνοριακών στοιχείων που χρησιμοποιείται για την επίλυση των προβλημάτων συνοριακών τιμών που αφορούν στον υπολογισμό των αξονικών, καμπτικών και στρεπτικών μορφών στρέβλωσης και διαστρέβλωσης και των γεωμετρικών σταθερών. Η δεύτερη μέθοδος είναι η μέθοδος των πεπερασμένων στοιχείων που χρησιμοποιείται για τη διακριτοποίηση των και την επίλυση των καθολικών εξισώσεων ισορροπίας της δοκού. Στη βάση των αναλυτικών και αριθμητικών μεθόδων που παρουσιάζονται στην παρούσα διδακτορική διατριβή έχουν γραφτεί προγράμματα σε FORTRAN 90/95 και έχουν μελετηθεί αντιπροσωπευτικά αριθμητικά παραδείγματα. Επιπρόσθετα, έχουν γραφτεί scripts σε MATLAB προκειμένου να ερμηνευθούν τα αποτελέσματα διαμέσου δισδιάστατων και τρισδιάστατων γραφημάτων της δοκού. Η ακρίβεια και η αξιοπιστία των προτεινόμενων μεθόδων επιβεβαιώνονται μέσω αριθμητικών παραδειγμάτων που αντλούνται από τη βιβλιογραφία και αποτελεσμάτων που εξάγονται από αναλύσεις τρισδιάστατων και επιφανειακών πεπερασμένων στοιχείων.

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