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The aim is at providing the reader with knowledge on both basic and advanced topics in the vibrations of structural systems categorized as either discrete or continuous. The emphasis is on numerical methods of solution with a Java programming framework that accompanies the book. Reference will be made to numerical solutions based on the contemporary discretization techniques ranging from the commonly used finite elements to more advanced mixed forms as well as boundary element methods. In terms of practical applications, the topic of modal analysis is of central importance in structural dynamics where response of the system is captured in the modal space through the modal characteristics, namely modal shapes and frequencies. Specifically, identification modal analysis which might be classified as experimental and/or operational modal analysis, focuses in the determining modal characteristics of a given structural system from a set of measured responses in the form of transient signals.Thus, vibration-based methods for damage detection and localization taking advantage of machine learning techniques are currently a topic that is seeing rapid development because of its relevance to structural health monitoring. Finally, in terms of numerical analysis, the book presents an open-source software platform named 'Symplegma'. This is developed using Java with the Apache-Groovy-Java virtual machine-based language serving as the scripting mechanism for adding special-purpose software modulus. This platform is presented in depth and provides a work space for the reader to implement the material presented in the book and to explore areas of application that are of interest to him/her.