Course detail

Dynamics III - Nonlinear and Stochastic Vibration

FSI-R3DAcad. year: 2025/2026

The students will have a basic knowledge of nonlinear and stochastic models of engineering systems and its operation and responses. They will be able to calculate a typical linearized model of these systems. They will be able to solve practical problems that can be modelled in this way.
The student will have knowledge of the chaotic operation and they will be able to analyse responses of dynamic system with random vibration or seismic excitation.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

doc. Ing. Zdeněk Hadaš, Ph.D.

Department

Institute of Solid Mechanics, Mechatronics and Biomechanics (ÚMTMB)

Entry knowledge

Linear algebra, differential equations, strength of materials, kinematics and dynamics of particles and bodies, solution of N degrees of freedom system oscillation, numerical methods of linear algebra and ODE solutions, MATLAB or Python programming.

Rules for evaluation and completion of the course

The course-unit credit is granted under the condition of active participation in seminars and gain at least 20 points of 40. The gained points from the exercise is part of the final classification of the subject.
Final examination: The exam is divided into two parts. The evaluation of the exam is based on the classifications of each part. If one of the parts is graded F, the final grade of the exam is F. The content of the first part is a test, of which a maximum of 20 points can be obtained. The content of the second part is a solution of typical presented problems. It is possible to gain up to 40 points from this part. The form of the exam, types, number of examples or questions and details of the evaluation will be given by the lecturer during the semester. The final evaluation is given by the sum of the points gained from the exercises and exam. To successfully complete the course, it is necessary to obtain at least 50 points, where the maximum of 100 ECTS points can be reached.


Attendance at practical training is obligatory. Head of seminars carry out continuous monitoring of student's presence, their activities and basic knowledge. One absence can be compensated for by elaboration of substitute tasks

Aims

The aim of the course is to acquaint students with the specifics of behavior of nonlinear dynamic models in both frequency and time domain. Students will learn basic methods of solution by linearization of nonlinear models and numerical solutions. Students will also gain an overview of chaotic behavior, attractors, bifurcation and fractals. The aim of the second part of the course is to acquaint students with the basics of stochastic mechanics and to solve dynamic problems of random excitation and seismic events.
Absolvent bude mít znalosti v oblasti modelování nelineárních technických systémů a bude znát odezvy a projevy nelineárních dynamických systémů. Bude schopen linearizovat dynamický systém v okolí pracovního bodu. Bude schopen řešit úlohy technické praxe, které mohou být modelovány tímto způsobem. Absolvent bude seznámen s problematikou chaotického chování a stochastické mechaniky. Bude schopen analyzovat odezvy soustavy zatížené seismickou událostí a při zatížení náhodným buzením.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Jinqiao Duan: An Introduction to Stochastic Dynamics (Cambridge Texts in Applied Mathematics) 1st Edition, 2015. (EN)
BREPTA, Rudolf, Ladislav PŮST a František TUREK, 1994. Mechanické kmitání. Technický. Praha: Sobotáles.  (CS)
Steven H. Strogatz: Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering (Studies in Nonlinearity) 1st Edition, CRC Press, 2000. (EN)

Recommended reading

Not applicable.

Classification of course in study plans

  • Programme N-IMB-P Master's

    specialization IME , 1 year of study, summer semester, compulsory
    specialization BIO , 1 year of study, summer semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

doc. Ing. Zdeněk Hadaš, Ph.D.

Syllabus

1. Introduction of nonlinear dynamics
2. Stability of dynamic systems
3. Linearizing at an operating point
4. Nonlinear models of engineering systems and its analysis
5. Bifurcation
6. Chaotic operation and attractors
7. Fractals
8. Introduction of stochastic mechanics
9. Random vibrations in time domain
10. Random vibrations in frequency domain
11. Assessment of structures with seismic event excitation
12. Response of system with random vibrations excitation
13. Assessment of fatigue limit state at random vibrations

Computer-assisted exercise

26 hod., compulsory

Teacher / Lecturer

doc. Ing. Zdeněk Hadaš, Ph.D.

Syllabus

- Linear vs. nonlinear models
- Stability of dynamic systems
- Linearizing at an operating point
- Numeric solution of nonlinear system response
- Phase diagrams and attractors
- Self excited oscillations
- Bifurcation and chaos
- Typical tasks of stochastic mechanics
- Random vibrations in time domain
- Random vibrations in frequency domain
- Response of structure with seismic event excitation
- Response of model with random vibrations excitation
- Fatigue limit state at random vibrations