Course detail


FAST-NDB018Acad. year: 2024/2025

Assessment of civil engineering structures subjected to dynamic loads. Vibration theory fundamentals. Free vibration of single degree of freedom systems (SDOF). Experimental determination of fundamental natural frequency and damping factor. Response of SDOF systems to harmonic excitation. Response of SDOF systems to special forms of excitation and to general dynamic excitation. Frequency domain analysis. DFT, FFT. Mathematical models of continuous systems - axial and transverse vibration of elastic beams. Vibration of thin flat plate. Mathematical models of multi degree of freedoms (MDOF) systems. Application of Newton’s Laws to lumped-parameter models. Hamilton’s principle. Lagrange’s equations. Application of Lagrange’s equations to continuous systems. Free vibration of MDOF systems. Dynamic response by mode superposition method. The eigenvalue problem and numerical evaluation of modes and frequencies of MDOF systems. Dynamic analysis by finite element method (FEM). Element stiffness, damping, mass matrices and element force vector. Assembly of system Matrices. Vibration analysis employing FEM models. Direct integration methods for dynamic response. Response of systems to seismic excitation.

Language of instruction


Number of ECTS credits


Mode of study

Not applicable.


Institute of Structural Mechanics (STM)

Entry knowledge

Diagrams of internal forces on a beam, the meaning of the quantities: stress, strain and displacement, Hook’s law, equilibrium conditions for a beam, physical and geometrical equations for a beam, foundations of higher mathematics, variational principles, theory of plates, theory of FEM,.

Rules for evaluation and completion of the course

Extent and forms are specified by guarantor’s regulation updated for every academic year.


To get knowledge from the structure vibration theory, acquire appropriate terminology, recognize advantages of the alternatives to the dynamic analysis models, utilize up-to-date solving methods. Skills can be used as a basis for the real design of dynamically loaded structures; the theoretical knowledge helps to understand dynamic analyses implemented in modern computational programs based on FEM.
The course output is knowledge in the theory of structure vibration area. Student knows necessary technicals and became acquainted with option of computational models useful for dynamic analyses of the structure. The skills are possible to apply in solving of dynamic response of the structure. Obtained knowledge and acquirements are the basis for practical design and appreciation of dynamically loaded structures. The theoretic knowledge is instrumental towards to understanding of single type of dynamic analyses implemented in modern computational programs based on FEM.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Not applicable.

Recommended reading

Not applicable.

Classification of course in study plans

  • Programme NPC-SIS Master's, 1. year of study, summer semester, compulsory-optional

Type of course unit



26 hours, optionally

Teacher / Lecturer


1. Assessment of civil engineering structures subjected by dynamic loads. 2. Foundations the theory vibration of civil engineering structures. Models with single degree of freedom system (SDOF). 3. Free Vibration. Response SDOF systems to specials form of excitation. Damping models. 4. Measurement of frequencies and damping. Response of SDOF to general type of action. 5. Numerical analysis of SDOF response. Frequency analysis. FFT. 6. Continuous computational models – tension and bending of beam. Modal analysis. Vibration of plates. 7. Newton law application. Hamilton principle. 8. Multi degree of freedom models. Lagrange equations. 9. Discrete and continuous models. Modal analysis of two degree of freedom models. 10. Response solution using mode superposition method. Rayleigh method. 11. Natural frequency and eigenvalue vectors characteristics. Rayleigh-Ritz method. General eigenvalues problem. 12. Dynamic analysis by finite element method (FEM). Element matrices. The global system of equations Systems matrices. Modal analysis. Direct integration equations of motion. 13. Response solution structures on seismic loads.


26 hours, compulsory

Teacher / Lecturer


1. Calculation of equivalent stiffness and mass of models with single degree of freedom system (SDOF). 2. Derivation of equation of motion of SDOF systems. 3. Free vibration of undamped SDOF system – calculation of natural frequencies. 4. Free vibration of undamped SDOF system – calculation of damping parameters. 5. Response of SDOF system to harmonic excitation. 6. Response of SDOF system to various type excitations (impulse, constant force etc.). 7. Calculation of frequencies and modes of vibrations of continuous systems – rods and plates. 8. Derivation of equation of motion system with 2DOF (translational and rotational motion). 9. Assembly equation of 2DOF systems to calculate the frequencies and modes of vibrations and their solution. 10. Assembly modal matrices. Using procedures for normalizing mode of vibration and plotting modes. 11. Solution by mode-superposition method of the 2DOF system to harmonic excitation. 12. Tuning dampers for vibration reduction simple systems. 13. Derive elastic response spectra for solutions to seismic excitation.