MECH 429 Mechanical Vibrations Eşref Eşkinat Topics • Introduction, modeling, harmonic motion • One-DOF Free Vibrations, Damping • Harmonically excited vibration , direct forcing, resonance • Base motion, rotating unbalance, force transmission • Periodic and non-periodic forcing • Vibration measurement and isolation • 2 DOF free and forced vibrations, coupling • Multi-DOF Systems, Lagrange equations • Eigenvalue problem, natural frequencies, mode shapes • Modal analysis • Vibration absorbers Vibration • It is an everyday phenomenon we meet on everyday life. • Vibration analysis plays an important role in manufacturing machines, engines, automobiles, planes etc. Vibration • Vibration can be utilized profitably in vibratory conveyors, hoppers, sieves, compactors, washing machines, vibratory testing of materials, electronic filter circuits, design of nuclear reactors, earthquake simulator, etc. • In metal cutting processes, design with poor vibration analysis can cause chatter, which leads to a poor surface finish. Basic concepts • Definition: Any motion that repeats itself with a period. • Examples : Pendulum, guitar string, machines with unbalance. • The study of vibration is concerned with the oscillatory motion of bodies and the forces associated with them. • Examples of oscillatory or vibratory systems include machines, automobiles, aircrafts space shuttles, structures, etc. • A vibratory system consists of a means for storing Potential Energy (spring or elasticity), a means for storing Kinetic Energy (mass or inertia) and a means by which energy is gradually lost (damper) • Mechanism of vibration : Transfer of Potential Energy to Kinetic Energy and vice versa. Importance • Study of vibration is motivated by its detrimental as well as useful effects in engineering applications. • An automobile having engine with low vibrations and low noise is preferred for comfort. • Design of structures requires vibration analysis. Improper design may cause sudden failures. Example: Collapse of Tacoma Narrows bridge (1940) by wind-induced vibration. • Vibration analysis leads to design of more efficient turbines, steam engines, gasoline engines due to savings in undue dissipation of energy. Videos • Bridge collapse: http://www.youtube.com/watch?v=j-zczJXSxnw • Helicopter resonance: http://www.youtube.com/watch?v=0FeXjhUEXlc Vibration parameters • Vibrating mechanical systems can be modeled by containing three basic components: Spring, Mass, Damper • When these components are subjected to constant force, they react with a constant displacement, velocity and acceleration Vibration Analysis PHYSICAL DYNAMIC SYSTEM MATHEMATICAL MODEL GOVERNING EQUATIONS SOLVE GOVERNING EQUATIONS INTERPRET THE RESULTS FBD 9 Classification Vibration can be classified in several ways. Some of the important classifications a) Free and forced vibration b) Undamped and damped vibration c) Linear and nonlinear vibrations d) Deterministic and random vibration The term “Free Vibration” is used for the study of natural vibration modes in the absence external loading. The term “Forced Vibration” is used for the study of motion as a result of loads that vary with time. Loads that vary with time are called dynamic loads.