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 , th e AdéiRAcadémie Royal ldSie des Sciences iPiin Paris off ffdered a pri ze 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 ical lI 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AcosmodttA cos  sin mod tt sin m

ma bav 2sinsinAttmod m

2 22 2 mmab2  mm ab2 mm ab  EAa cosmod t A sin mod t  A 22  cos 2 1 t mmm 22 mmaa2 EAb 2sin2mod tA 1cos 2 1  t mm

22 22mmab mm ab EAa 22 2cos A21  t mm mm22 EA22cos22aa A  t b mm2221

mm  m EE A2221cos Aab a   t ab m2 21 mod  0.05

av 1 mmab1 mmab1, 5 mmab 5,  1 FdVibiForced Vibration of the Tide Derive the un -damped vibration resonance

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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 tan1 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 termi ne the amp litu de o f s tea dy-stttate vib ibtiration of ff 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