The authors investigate regular periodic structures consisting of repeated elements in the presence of translational symmetry and additional geometric symmetry. Such structures arise in many areas of natural sciences. The computational models of real-life engineering structures are hierarchical and depend on many parameters. Therefore, the main problem elaborated in this monograph is the dimensional reduction of models using the group representation theory.
The introductory Chapter 2 describes mechanical vibratory systems with hierarchical structure in general together with simulation methods.
Two most widely used approaches to the mathematical simulation of such structures are: \(1^{\circ}\). Methods using lumped parameter models, i.e., the discretization of the original system and its representation as the system with lumped parameters, often at the usage of finite elements method (FEM). \(2^{\circ}\). The usage of idealized elements with distributed parameters and known analytical solutions, both for the local elements and for subsystems.
Further the book is divided into two parts. The first one is devoted to mechanical systems with lumped parameters and contains two chapters. Chapter 3, “Vibrations of regular systems with periodic structure”, deals with vibrations of frames with periodic structure; dynamic properties of laminated systems; FEM for beam systems; hierarchy of mathematical models; superposition of wave motions; vibrations of self-similar structures in mechanics. Chapter 4, “Vibrations of systems with geometric symmetries, quasi-symmetrical systems”, gives basic information about the theory of group representation (TGR); application of TGR to mechanical systems: generalized projection operators of symmetry; vibrations of frames with cyclic symmetry; effects of FE mesh on matrix structure; quasi-symmetrical systems; hierarchy of symmetries, multiplication of symmetries; periodic systems consisting of symmetric elements; generalized modes in planetary gear reduction due to symmetry.
The second part deals with systems with distributed parameters and consists of Chapters 5–11: Chapter 5 “Basic equations and numerical methods”; Chapter 6 “Systems with periodic structure”; Chapter 7 “Systems with cyclic symmetry”; Chapter 8 “Systems with reflection symmetry elements”; Chapter 9 “Self-similar structures”; Chapter 10 “Vibrations of rotor systems with periodic structure”, and Chapter 11 “Vibrations of regular ribbed cylindrical shells”.
Appendices A, B, C and D provide stiffness and inertia matrices for four types of mechanical systems.