Elastic and Thermoelastic Problems in Nonlinear Dynamics of Structural Members: Applications of the Bubnov-Galerkin and Finite Difference Methods

ISBN-13: 9783030376659 / Angielski / Miękka / 2021 / 602 str.

Jan Awrejcewicz; Vadim a. Krysko

Spis treści

Second/New Edition (in bold the new material):

1 Introduction (to be updated).- 2 Coupled Thermoelasticity and Transonic Gas Flow.- 2.1 Coupled Linear Thermoelasticity of Shallow Shells.- 2.1.1 Fundamental Assumptions.- 2.1.2 Differential Equations.- 2.1.3 Boundary and Initial Conditions.- 2.1.4 An Abstract Coupled Problem.- 2.1.5 Existence and Uniqueness of Solutions of Thermoelasticity Problems.- 2.2 Cylindrical Panel Within Transonic Gas Flow.- 2.2.1 Statement and Solution of the Problem.- 2.2.2 Stable Vibrating Panel Within a Transonic Flow.- 2.2.3 Stability Loss of Panel Within Transonic Flow.- 3 Estimation of the Errors of the Bubnov—Galerkin Method.- 3.1 An Abstract Coupled Problem.- 3.2 Coupled Thermoelastic Problem Within the Kirchhoff-Love Model.- 3.3 Case of a Simply Supported Plate Within the Kirchhoff Model.- 3.4 Coupled Problem of Thermoelasticity Within a Timoshenko-Type Model.- 4 Numerical Investigations of the Errors of the Bubnov—Galerkin Method.- 4.1 Vibration of a Transversely Loaded Plate.- 4.2 Vibration of a Plate with an Imperfection in the Form of a Deflection.- 4.3 Vibration of a Plate with a Given Variable Deflection Change.- 5 Coupled Nonlinear Thermoelastic Problems.- 5.1 Fundamental Relations and Assumptions.- 5.2 Differential Equations.- 5.3 Boundary and Initial Conditions.- 5.4 On the Existence and Uniqueness of a Solution.- 6 Theory with Physical Nonlinearities and Coupling.- 6.1 Fundamental Assumptions and Relations.- 6.2 Variational Equations of Physically Nonlinear Coupled Problems.- 6.3 Equations in Terms of Displacements.- 7 Nonlinear Problems of Hybrid-Form Equations.- 7.1 Method of Solution for Nonlinear Coupled Problems.- 7.2 Relaxation Method.- 7.3 Numerical Investigations and Reliability of the Results Obtained.- 7.4 Vibration of Isolated Shell Subjected to Impulse.- 7.5 Dynamic Stability of Shells Under Thermal Shock.- 7.6 Influence of Coupling and Rotational Inertia on Stability.- 7.7 Numerical Tests.- 7.8 Influence of Damping e and Excitation Amplitude A.- 7.9 Spatial-Temporal Symmetric Chaos.- 7.10 Dissipative Nonsymmetric Oscillations.- 7.11 Solitary Waves.- 8 Dynamics of Thin Elasto-Plastic Shells.- 8.1 Fundamental Relations.- 8.2 Method of Solution.- 8.3 Oscillations and Stability of Elasto-Plastic Shells.- 9 Mathematical Model of Cylindrical/Spherical Shell Vibrations.- 9.1. Fundamental Relations and Assumptions. - 9.2. The Bubnov-Galerkin Method.- 9.2.1. Closed Cylindrical Shell.- 9.2.2. Cylindrical Panel.-  9.3. Reliability of the Obtained Results.- 9.4. On the Set up Method in the Theory of Flexible Shallow Shells.- 9.5. Dynamic Stability Loss of the Shells Under the Step-Type Function.- 10 Chaotic Vibrations of Cylindrical and Spherical Shells.- 10.1. Novel Models of Scenarios of Transition from Periodic to Chaotic Orbits.- 10.2. Sharkovskiy’s Periodicity Exhibited by PDEs Governing Dynamics of Flexible Shells.- 10.3. On the Space-Temporal Chaos.- 11 Mathematical Models of Chaotic Vibrations of Closed Cylindrical Shells with Circular Cross Section.- 11.1. On the Convergence of the Bubnov-Galerkin (BG) Method in the Case of Chaotic Vibrations of Closed Cylindrical Shells.- 11.2. Chaotic Vibrations of Closed Cylindrical Shells Versus Their Geometric Parameters and the Area of the External Load Action.- 12 Chaotic Dynamics of Flexible Closed Cylindrical Nanoshells under Local Load.- 12.1. Statement of the Problem.- 12.2. Algorithm of the Bubnov-Galerkin Method.- 12.3. Numerical Experiment.- 13 Contact Interaction of Two Rectangular Plates Made From Different Materials Taking into Account Physical Nonlinearity.- 13.1. Statement of the Problem.- 13.2. Reduction of PDEs to ODEs.- 13.2.1. Method of Kantorovich-Vlasov (MKV).- 13.2.3. Method of Variational Iteration (MVI).- 13.2.4. Method of Arganovskiy-Baglay-Smirnov (MABS).- 13.2.5. Combined Method (MC).- 13.2.6. Matching of the Methods of Kantorovich-Vlasov and Arganovskiy-Baglay-Smirnov (MKV+MABS).- 13.2.7. Matching of the Methods of Vaindiner and the Arganovskiy-Baglay-Smirnov (MV+MABS).- 13.2.8. Matching of the Methods of Vaindiner and the Method of Variational Iterations (MV+MVI).- 13.2.9. Numerical Example.- 13.3. Mathematical Background.- 13.3.1. Theorems on Convergence of MVI.- 13.3.2. Convergence Theorem.- 13.4. Contact Interaction of Two Square Plates.- 13.4.1. Computational examples.- 13.5. Dynamics of a Contact Interaction.- 14 Chaotic Vibrations of Flexible Shallow Axially Symmetric Shells vs. Different Boundary Conditions.- 14.1. Problem Statement and the Method of Ssolution.- 14.2. Quantification of True Chaotic Vibrations.- 14.3. Modes of Vibrations (Simple Support).- 14.4. Modes of Vibrations (Rigid Clamping).- 14.5. Investigation on the Occurrence of Ribs (Simple Nonmovable Shell Support).- 14.6. Shell Vibration Modes (Movable Clamping).- 15 Chaotic Vibrations of Two Euler-Bernoulli Beams with a Small Clearance.- 15.1. Mathematical Model.- 15.2. Principal Component Analysis (PCA).- 15.3. Numerical Experiment.- 15.4. Application of the Principal Component Analysis.- 15.5. Concluding Remarks.- 16 Unsolved Problems in Nonlinear Dynamics of Shells.- References.- Index.

O autorze

From the beginning of his academic career Jan Awrejcewicz has been associated with the Mechanical Faculty of the Lodz University of Technology, where he obtained a master's degree in engineering, a PhD in technical sciences and a postdoctoral degree (habilitatioin). In 1994, he received the title of Professor from the President of Poland. In 1998 he founded the Department of Automation, Biomechanics and Mechatronics, that he is still managing. Since 2013 he has been a member of the Polish Central Commission for Degrees and Titles, and since 2019 also of the Council for Scientific Excellence. He is also an Editor-in-Chief of 3 international journals and member of the Editorial Boards of 90 international journals (23 with IF) as well as editor of 25 books and 28 journal special issues. He also reviewed 45 monographs and textbooks and over 600 journal papers for about of 140 journals. His scientific achievements cover issues related to asymptotic methods for continuous and discrete mechanical systems, taking into account thermoelasticity and tribology, and computer implementations using symbolic calculus, nonlinear dynamics of mechanical systems with friction and impacts, as well as engineering biomechanics. He authored/co-authored over 780 journal papers and refereed international conference papers and 50 monographs. For his scientific merits he recived numerous prestigious awards and distinctions, among them titles of the Honoratry Doctor of Częstochowa University of Technology (2013), University of Technology and Humanities in Bielsko-Biała (2013), Kielce University of Technology (2019), National Technical University "Kharkiv Polytechnic Institute" (2019), and Gdańsk University of Technology (2019).

Vadim Krysko was born in Kiev, Ukraine, on September 21, 1937. He received a master's degree in Civil Engineering from the Saratov State Technical University in 1962. A Ph.D. degree in Mechanics of Solids he obtained from the Saratov State Technical University, USSR in 1967, and D.Sci. degree in Mechanics of Solids he obtained from Moscow Civil Engineering University in 1978. In 1982 he became a full professor, the academic rank of professor obtained from the Department of Higher Mathematics, Saratov State Technical University. He is author/co-author of 329 publications in scientific journals and conference proceedings, 8 monographs in English, 8 monographs in Polish, and 16 monographs in Russian. Topics of his research cover such branches of mechanics as thermoelasticity and thermoplasticity, the theory of optimization of mechanical systems, the theory of propagation of elastic waves upon impact, the theory of coupled problems of thermoelasticity and the interaction of flexible elastic shells with a transonic gas flow, numerical methods for solving nonlinear problems of shells theory. Vadim Krysko was a promoter of 58 PhD (postgraduate student) theses. He is currently the Head of the Department of Mathematics and Modeling, Saratov State Technical University, Russia.  In 2012 he was honored by a title of Doctor Honoris Causa of the Lodz University of Technology, Poland. Vadim Krysko opened and developed novel scientific directions for research in the construction, justification and numerical implementation of new classes of equations of mathematical physics of hyperbolic-parabolic types and proposed effective methods for their numerical solution.

 

 

Blurb

Adding seven new chapters integrating modern approaches to dynamical systems theory, this revised second edition describes numerical approaches to the nonlinear theory of beams, plates and shells, with an emphasis to the elastic and thermoelastic features. The added chapters discuss in particular how to understand, model, and obtain reliable results while detecting and analysing chaotic dynamics exhibited by thin structural members. The authors attempt to address the large impact of new directions of nonlinear dynamical phenomena in the context of the overall state-of-the art presented in the first book edition. They also present new perspectives of the unsolved problems of thin structural members.

The authors made every effort to keep the text intelligible for both practitioners and graduate students, although they offer a rigorous treatment of both purely mathematical and numerical approaches presented so that the reader can understand, analyse and track the nonlinear dynamics of spatial systems (shells) with thermomechanical behaviours.