Mechanics 2

2022/2023
Programme:
Mathematics, First Cycle
Year:
3 year
Semester:
second
Kind:
optional
Group:
B1
ECTS:
5
Language:
slovenian
Hours per week – 2. semester:
Lectures
2
Seminar
0
Tutorial
2
Lab
0
Prerequisites

Completed courses Analysis 2a and Analysis 2b.

Content (Syllabus outline)

• Small oscillations. Lagrange equations. Second order approximation. Normal modes and normal coordinates. The continuum limit.
• Stability. Linear stability, Ljapunov stability. Poincare Bendixson theorem. Introduction to bifurcation theory. Euler buckling.
• Perturbation methods. Perturbation of algebraic equations, Kepler equation. Perturbation of differential equations, anharmonic oscillation. Van der Pol oscillator. Singular perturbation. Boundary layer. Multiple time scales.
• Variational methods in mechanics. Hamilton's principle. Variational principles in elastostatics. Finite element method.

Readings

• Cassel, K.W. Variational Methods with Applications in Science and Engineering, Cambridge, 2013.
• Dym, C.L., Stability theory and Its Applications to Structural Mechanics, Dover, 2002.
• Holmes, M.H., Introduction to Perturbation Methods, Springer, 2013.
• Goldstein, H., C.P. Poole, J.L. Safko, Classical Mechanics, 2001.
• Križanič, F. Navadne diferencialne enačbe in variacijski račun, Državna založba, Ljubljana, 1974.

Objectives and competences

Presentation of some mathematical methods of classical mechanics and their contemporary application to some selected problems in mechanics.

Intended learning outcomes

• Knowledge and understanding: Students acquire familiarity with some mathematical methods to solve nonlinear problems. They also get understanding of basic principles of qualitative analysis and variational methods.
• Application: solving and qualitative analysis of some typical problems from natural science.
• Reflection: Understanding of the theory and the methods through solving some typical problems.
• Transferable skills: To enhance knowledge and understanding of mathematical methods for solving problems from natural science and technology.

Learning and teaching methods

Lectures, exercises, usage of computer algebra, homework, consultation.

Assessment

Two midterm exams or alternatively a written exam: 50.%
Project work with an oral exam: 50%
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

MEJAK, George. Variational formulation of the equivalent eigenstrain method with an application to a problem with radial eigenstrains. International journal of solids and structures, ISSN 0020-7683. [Print ed.], 2014, vol. 51, iss. 7-8, str. 1601-1616. [COBISS-SI-ID 17128281]
MEJAK, George. Eshebly tensors for a finite spherical domain with an axisymmetric inclusion. European journal of mechanics. A, Solids, ISSN 0997-7538. [Print ed.], 2011, vol. 30, iss. 4, str. 477-490. [COBISS-SI-ID 16025177]
MEJAK, George. Finite element solution of a model free surface problem by the optimal shape design approach. International journal for numerical methods in engineering, ISSN 0029-5981. [Print ed.], 1997, vol. 40, str. 1525-1550. [COBISS-SI-ID 9983833]

KOC, Pino. Sea-wave dynamic loading of sailing yacht`s retractable keel. Strojniški vestnik, ISSN 0039-2480, Mar. 2014, vol. 60, no. 3, str. 203-209, ilustr., doi: 10.5545/sv-jme.2013.1423. [COBISS-SI-ID 13401627]
KOC, Pino, HALILOVIČ, Miroslav, ŠTOK, Boris. Impact of restrained thermal expansion on NPP Krško primary loop piping. Tehnički vjesnik, ISSN 1330-3651, 2013, god. 20, br. 5, str. 897-904, ilustr. [COBISS-SI-ID 13212955]
KOC, Pino, ŠTOK, Boris. Computer-aided identification of the yield curve of a sheet metal after onset of necking. Computational materials science, ISSN 0927-0256. [Print ed.], 2004, letn. 31, št. 1/2, str. 155-168. [COBISS-SI-ID 7467803]