Debussy: La Mer Galileo Galilei in his 1632 Dialogue Concerning the Two Chief World Systems, whose working title was Dialogue on the Tides, attributed the tides to the sloshing of water caused by the Earth's movement around the sun. At the same time Johannes Kepler correctly suggested that the Moon caused the tides. Isaac Newton was the first person to explain tides, in the Principia (1687), as the product of the gravitational attraction of astronomical masses. In 1740, the AdéiRAcadémie Roya ldSile des Sciences iPiin Paris o ffdffered a pr ize f or th e best theoretical essay on tides. Daniel Bernoulli, Leonhard Euler, Colin Maclaurin and Antoine Cavalleri shared the prize. Pierre-Simon Laplace formulated the first major dynamic theory for water tides. The Laplace tidal equations are still in use today. Kelvin and Henri Poincaré further developed Laplace's theory. EN4 : Vibrations March, 2013 Lectured by K.-S. Kim 3/19 Lecture 14: Damped Free Vibrations: Transient response and Damping criticality Car Suspensions and Atomic Force Microscope 3/21 Lecture 15: Forced Vibrations: Harmonic Excitation and Resonance OTidOcean Tide andMd Mus ica lIl Instruments 3/26 and 3/28 Spring Break 4/2 Lecture 16: Forced Vibrations: Magnification Factor and Phase Shift History of Suspension and Vibration Isolation HW 5-5.5: Vibration-Energy Transfer trough Weak Coupling DfiDefine themodltidulation frequency mod 2 1 2 and the average frequency av 21 2 mmab aavav AcosmodttA cos sin mod tt sin m ma bav 2sinsinAttmod m 2 22 2 mmab2 mm ab2 mm ab EAa cosmod t A sin mod t A 22 cos 2 1 t mmm 22 mmaa2 EAb 2sin2mod tA 1cos 2 1 t mm 22 22mmab mm ab EAa 22 2cos A21 t mm mm22 EA22cos22aa A t b mm2221 22mmab m a EEab A21cos A 21 t m2 mod 0.05 av 1 mmab1 mmab1, 5 mmab 5, 1 FdVibiForced Vibration of the Tide Derive the un -damped vibration resonance *** Desirable resonance: Musical instruments Sensor signal amplification Undesirable resonance: Tacoma bridge collapse Voice shatteringgg of wine glass Steady-state solution – external force L0 x(t) k, L F(t)=F0 sin t 2 0 12dx dx Ft () kc m x 22 dt k n , n dt n m 2 km xt() X0 sin t Fk 2/ X 0 tan1 n 0 1/2 2 2 222 2 1/ n 1/nn 2/ System vibrates at same frequency as force Amplitude depends on forcing frequency, nat frequency, and damping coeft. Forced Vibrations – concept checklist You should be able to: 1.Be able to derive equations of motion for spring-mass systems subjected to external forcing (several types) and solve EOM using complex vars, or byypg comparing to solution tables 2.Understand (qualitatively) meaning of ‘transient’ and ‘steady-state’ response of a forced vibration system (see Java simulation on web) 3.Understand the meaning of ‘Amplitude’ and ‘phase’ of steady-state response of a forced vibration system 4.Understand amplitude-v-frequency formulas (or graphs), resonance, high and low frequency response for 3 systems 5.De term ine the amp litu de o f s tea dy-stttate vib ibtiration o fff forced spri ng-mass systems. 6.Deduce damping coefficient and natural frequency from measured forced response of a vibrating system 7.Use forced vibration concepts to design engineering systems .