• Semester 6, position 4

It is necessary to enable the students to independently form and solve linear differential equations of motion of mechanical models of real objects oscillatory moving in different areas of mechanical engineering.

Upon successful completion of this course, students will be able to: • Determine equilibrium position of conservative mechanical system with finite number of degrees of freedom. • Form differential equations of motions of small mechanical vibrations of a mechanical system about the equilibrium position in matrix form (determine generalized mass, stiffness and damping matrices, as well as vector of generalized forces transformed on Fourier series). • Analyze free and forced, as well as damped and undamped linear mechanical vibrations, in a clear observation of phenomena in linear mechanical vibration as well as resonance, beating and the dynamic absorber). • Calculate (analytical and numerical) quantities which characterize vibration processes: natural frequencies, amplitudes, phase angles, logarithmic decrements and modal matrix. • Determine equations of motion in analytical form using software (Matlab...) for systems with large number of degrees of freedom. • Describe free undamped mechanical vibrations of elastic bodies with 1-D mass distribution with appropriate partial differential equations, for cases of longitudinal, torsion and lateral vibrations. • Numerically solve characteristic equation for various cases of boundary conditions and determine angular frequencies. Determine analytical solutions of appropriate partial differential equations in simpler cases initial and boundary conditions.

Stability of equilibrium of the conservative system. Silvester's criteria. Linearization of the differential equations of motion. Vibration of the conservative system. Frequencies. The main mode shapes of vibration. Modal matrix. Conservative systems with special values of natural frequencies (eigenvalues). Vibration of the body on the beam supports. Damped vibration. Forced undamped vibration. Forced vibration. Resonance. Beating. Dynamic amplification factor. The dynamic absorber without damping. Linear oscillations of non-stationary system. Forced damped vibration of the system. Lateral vibration of string. Longitudinal vibration of prismatic bodies. Torsional vibration of the shaft with circular cross section. Lateral vibration of prismatic bodies.

Stability of equilibrium of the conservative system. Silvester's criteria. Linearization of the differential equations of motion. Vibration of the conservative system. Frequencies. The main mode shapes of vibration. Modal matrix. Conservative systems with special values of natural frequencies (eigenvalues). Vibration of the body on the beam supports. Damped vibration. Forced undamped vibration. Forced vibration. Resonance. Beating. Dynamic amplification factor. The dynamic absorber without damping. Linear oscillations of non-stationary system. Forced damped vibration of the system. Lateral vibration of string. Longitudinal vibration of prismatic bodies. Torsional vibration of the shaft with circular cross section. Lateral vibration of prismatic bodies.

The subject can take students who have made a condition for entry into the third year of study.

Vuković, J., Obradović,A., Linear vibrations of mechanical systems, Mašinski fakultet, Beograd, 2021., handouts Ružić D., Čukić R., Dunjić M., Milovančević M., Anđelić N., Milošević-Mitić V.: Strength of Materials,Book 5, Tables, Mašinski Fakultet, Beograd 2007. Lazić D., Ristanović M.: Introduction to MATLAB , Mašinski fakultet, Beograd 2005. MATLAB software

Total assigned hours: 75

New material: 20
Elaboration and examples (recapitulation): 10

Auditory exercises: 24
Laboratory exercises: 6
Calculation tasks: 0
Seminar paper: 0
Project: 0
Consultations: 0
Discussion/workshop: 0
Research study work: 0

Review and grading of calculation tasks: 0
Review and grading of lab reports: 4
Review and grading of seminar papers: 0
Review and grading of the project: 0
Test: 6
Test: 0
Final exam: 5

Activity during lectures: 0
Test/test: 45
Laboratory practice: 15
Calculation tasks: 0
Seminar paper: 0
Project: 0
Final exam: 40
Requirement for taking the exam (required number of points): 30

Rao S.S.: Mechanical vibrations, Addison-Wesley Publishing Company Inc., 1995.; Vujanović B.: Theory of vibrations, Fakultet tehničkih nauka, Novi Sad 1995. ; Kojić M., Mićunović M.: Theory of vibrations, Naučna knjiga, Beograd 1991. ; Vujičić V.: Theory of vibrations, Naučna knjiga, Beograd 1977.