Mechanics 2

Completed courses Analysis 2a and Analysis 2b.

• 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.

1. K. W. Cassel: Variational methods with applications in science and engineering, Cambridge [etc.] : Cambridge University Press, 2013.
2. C. L. Dym: Stability theory and its applications to structural mechanics, Reprint, Mineola, N.Y. : Dover Publications, 2002.
3. H. Goldstein, C. Poole, J. Safko: Classical mechanics, 3rd ed., San Francisco : Addison Wesley, 2001.
4. M. H. Holmes: Introduction to perturbation methods, New York : Springer, 1995.
5. F. Križanič: Navadne diferencialne enačbe in variacijski račun, Ljubljana : Mladinska knjiga, 1974.
6. V. Moretti: Analytical mechanics : classical, Lagrangian and Hamiltonian mechanics, stability theory, special relativity, Cham : Springer, cop. 2023.

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

• 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.

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

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)

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]