Single-Course Danish 5 ECTS

Dynamics and vibrations

Overall Course Objectives

Mechanical structures and mechanisms often operate under many forms of dynamic/vibrating/oscillating loading, or work by exerting vibrations on other structures or processes; think of parts in cars, airplanes, turbines, pumps, wind turbines, percussion drillers, music instruments, etc. etc. Mathematical modeling and analysis is decisive for predicting, limiting, eliminating, amplifying, controlling, or just understanding such influences and processes, and thus also for mechanical design and troubleshooting. This course provides a fundamental theoretical background for solving vibration problems related to mechanical structures and machinery, and for advanced studies in dynamics and vibrations.

Identify sources for inertia, stiffness / restoring force, energy-dissipation, and external loads for mechanical systems.
Identify relevant degrees of freedom for mathematical modeling mechanical systems.
Use Newton laws and free body diagrams to determine the equations of motion for simple models of mechanical systems with a finite or infinite number of degrees of freedom (i.e. discrete or continuous structures).
Use Lagrange’s equations to determine the equations of motion for simple models of mechanical systems with a finite number of degrees of freedom.
Rewrite equations of motion for specific models into the standard form of ordinary differential equations (scalar or matrix-vector form), or scalar partial differential equations.
Use mathematical and numerical methods for solving standard equations of motion for models of mechanical systems.
Give practically useful interpretations and evaluations of analytical and numerical results.
Identify resonance problems for mechanical systems whose dynamic characteristics (i.e. inertia and energy dissipation) cannot be neglected.
Apply common techniques for quenching or damping unwanted mechanical vibrations.
Account for the limitations in the models and methods used, and predict the possible consequences of making simplified assumptions, especially linearization and limitation of the number of degrees of freedom.

1. Application of kinematical and kinetic theory (incl. Newton’s laws) for setting up equations of motions for vibrating mechanical systems.
2. Vibrations of linear single-degree-of-freedom systems: Free and forced vibration; damping.
3. Vibrations of linear multiple-degree-of-freedom systems: Equations of motion; Lagrange’s equations; Modal analysis; Rayleigh’s quotient and method; Vibration damping.
4. Vibrations of linear continuous systems: Equations of motion; Rayleigh’s quotient and method; Transverse vibration of strings; Axial and torsional vibrations of rods; Introduction to transverse vibrations of beams.
5. Introduction to nonlinear vibration analysis.

41501/41015/01035

Lectures, demo experiments, problem solving