This book is a textbook on mechanical vibrations, intended for M.Sc. students in mechanical engineering, both Hungarian and foreign. It is based on lectures given to students who must take the course on mechanical vibrations. The authors faced the challenge of determining the level of preliminary knowledge expected from students, as Hungarian students have some familiarity with tensor algebra and analysis, while foreign students may not. To address this, the book uses matrix notations primarily, with some sections using vector and tensor notations for more advanced topics. The first chapter serves as an introduction to the fundamentals of dynamics, covering kinematic relations, effective forces, impulse and momentum, and work and energy. The text includes exercises and problems for students to solve independently. Chapter 2 discusses impact, including central and eccentric impacts, with graphical solutions and examples. Chapter 3 covers simple vibration problems, including undamped and damped free vibrations, forced vibrations, and applications such as machine foundations. Chapter 4 introduces multidegree of freedom systems, deriving Lagrange's equations and discussing spring-mass systems. Chapter 5 presents the general theory of multidegree of freedom systems, including eigenvalue problems and the Rayleigh quotient. Chapter 6 discusses rotating motion, including flywheels, shaft stability, and gyroscopic effects. Chapter 7 deals with systems with infinite degrees of freedom, covering longitudinal, transverse, and torsional vibrations. Chapter 8 focuses on eigenvalue problems of ordinary differential equations, introducing Green functions and their applications. Chapter 9 discusses eigenvalue problems of differential equation systems, including Timoshenko beams. Chapter 10 addresses eigenvalue problems of degenerated differential equation systems, including heterogeneous curved beams. Appendices provide an introduction to tensor algebra, useful relationships, and solutions to selected problems. The book is structured with chapters, sections, and appendices, and includes references and an index.This book is a textbook on mechanical vibrations, intended for M.Sc. students in mechanical engineering, both Hungarian and foreign. It is based on lectures given to students who must take the course on mechanical vibrations. The authors faced the challenge of determining the level of preliminary knowledge expected from students, as Hungarian students have some familiarity with tensor algebra and analysis, while foreign students may not. To address this, the book uses matrix notations primarily, with some sections using vector and tensor notations for more advanced topics. The first chapter serves as an introduction to the fundamentals of dynamics, covering kinematic relations, effective forces, impulse and momentum, and work and energy. The text includes exercises and problems for students to solve independently. Chapter 2 discusses impact, including central and eccentric impacts, with graphical solutions and examples. Chapter 3 covers simple vibration problems, including undamped and damped free vibrations, forced vibrations, and applications such as machine foundations. Chapter 4 introduces multidegree of freedom systems, deriving Lagrange's equations and discussing spring-mass systems. Chapter 5 presents the general theory of multidegree of freedom systems, including eigenvalue problems and the Rayleigh quotient. Chapter 6 discusses rotating motion, including flywheels, shaft stability, and gyroscopic effects. Chapter 7 deals with systems with infinite degrees of freedom, covering longitudinal, transverse, and torsional vibrations. Chapter 8 focuses on eigenvalue problems of ordinary differential equations, introducing Green functions and their applications. Chapter 9 discusses eigenvalue problems of differential equation systems, including Timoshenko beams. Chapter 10 addresses eigenvalue problems of degenerated differential equation systems, including heterogeneous curved beams. Appendices provide an introduction to tensor algebra, useful relationships, and solutions to selected problems. The book is structured with chapters, sections, and appendices, and includes references and an index.