Harmonic excitation (damped)

Transcription

1 Harmoic eciaio damped k m cos EOM: m&& c& k cos c && ζ & f cos The respose soluio ca be separaed io par;. Homogeeous soluio h. Paricular soluio p h p & ζ & && ζ & f cos

2 Homogeeous soluio Homogeeous soluio h Same as free vibraio Uder damped < ζ < h ζ Ae si φ d where d ζ Over damped < ζ h A e ζ ζ A e ζ ζ Criically damped ζ h A A e

3 rom EOM p Paricular soluio && ζ & f cos The paricular soluio p ca be wrie i he form: X cos θ or A cos B si p s Subsiuio p io EOM, he coefficies X ad θ A s ad B s ca be deermied. The he paricular soluio is s p f ζ cos a ζ X θ

4 Respose of harmoic eciaio Respose of harmoic eciaio h p E Uderdamped Deped o ζ ζ Ae si φ X cos θ d h p Decrease wih ime Ampliude X is cosa Trasie respose Seady-sae respose

5 Respose of harmoic eciaio E Uderdamped ζ Ae si φ X cos θ d h p As, rasie respose dies ou ad oal respose p

6 Seady-sae respose rom seady-sae respose p f ζ cos a ζ Xk X ζr, where θ a f r ζr r r 4 X f Ampliude respose 8º θr Phase respose 9º r º r

7 Seady-sae respose rom seady-sae respose p f ζ cos a ζ Xk X f r ζr where r 4 X f Ampliude respose Maimum ampliude occurs whe d dr r ζr r ζ or ζ r Xk ma ζ ζ

8 requecy respose mehod Euler s formula EOM EOM comple form e j m&& cos j si c& k cos mz && cz& kz e j Real par of he comple soluio correspods o he soluio of EOM. p Re z where Subsiuig z io EOM comple form. z Ze j Z is a comple-valued cosa j m jc k Ze e j

9 requecy respose mehod j j e Ze k jc m j H j c m k Z comple frequecy respose fucio Hj θ j e c m k Z ] [ a θ m k c, where Re z p ] [ θ j e c m k z cos ] [ θ c m k p

10 requecy respose mehod 3 cos θ [ k m c ] p ; θ a c k m or p f cos θ ζ ; θ a ζ p i he form of frequecy respose fucio Z H j H j e jθ z H j e j θ p Re z H j cos θ p

11 Eample The block of mass m 45 kg is suspeded by wo sprigs each of siffess k 3 kn/m ad is aced upo by he force 35cos5 N where is he ime i secods. Deermie he ampliude X of he seadysae moio if he viscous dampig coefficie c is a ad b 9 Ns/m. [J. L. Meriam & L. G. Kraige 8/5]

12 Eample or a vibraig sysem, m kg, k 5 N/m, ad c 45 Ns/m. A harmoic force of ampliude 8 N ad frequecy 3.5 Hz acs o he mass. If he iiial displaceme ad velociy of he mass are 5 mm ad 5m/s, fid he complee soluio represeig he moio of he mass. [Sigiresu S. Rao, Mechaical Vibraios 4 h ediio i SI uis 3/33]

13 Eample 3 A machie par of mass.95 kg vibraes i a viscous medium. Deermie he dampig coefficie whe a harmoic eciig force of 4.46 N resuls i a resoa ampliude of.7 cm wih a period of. s. [William T. Thomso & Marie Dillo Dahleh, Theory of Vibraio wih Applicaios 5 h ediio 3/]

14 Eample 4 A weigh aached o a sprig of siffess 55 N/m has a viscous dampig device. Whe he weigh is displaced ad released, he period of vibraio is.8 s, ad he raio of cosecuive ampliudes is 4. o.. Deermie he ampliude ad phase whe a force cos3 acs o he sysem. [William T. Thomso & Marie Dillo Dahleh, Theory of Vibraio wih Applicaios 5 h ediio 3/3]

15 Eample 5 A sprig-mass is ecied by a force cos. A resoace, he ampliude is measured o be.58 cm. A.8 resoa frequecy, he ampliude is measured o be.46 cm. Deermie he dampig facor ζ of he sysem. [William T. Thomso & Marie Dillo Dahleh, Theory of Vibraio wih Applicaios 5 h ediio 3/5]