Module: 8 Lecture: 1

Transcription

1 Moule: 8 Lecture: 1 Energy iipate by amping Uually amping i preent in all ocillatory ytem. It effect i to remove energy from the ytem. Energy in a vibrating ytem i either iipate into heat oun or raiate away. In vibration analyi, we are generally concerne with amping in term of ytem repone. The lo of energy from the ocillatory ytem reult in the ecay of amplitue of free vibration. In teay tate force vibration, the lo of energy i balance by the energy which i upplie by the excitation. A vibrating ytem may encounter many ifferent type of amping force, from internal molecular friction to liing friction an flui reitance. Generally their mathematical ecription i quite complicate an not uitable for vibration analyi. Thu implifie amping moel have been evelope that in many cae are not foun to be aequate in evaluating the ytem repone. Energy iipate i uually etermine uner the conition of cyclic ocillation. Depening on the type of amping preent, the force-iplacement relationhip when plotte may iffer greatly. In all cae, however, the force-iplacement curve will encloe an area, referre to a the hyterei loop that i proportional to the energy lo per cycle. The energy lot per cycle ue to a amping force i compute from the general Hyterei loop x Area = energy lo per cycle Page1

2 equation W x c W c c We conier in thi ection the implet cae of energy iipation, that of a pring-ma ytem with vicou amping. The amping force in thi cae i c. With the teay tate iplacement an velocity x in t x in t x co t the energy iipate per cycle become W c x c c co t c c In a imilar line work one by pring an inertia force can be written a follow: W W I k x x k in I x x m x x 3 m t cot in m kx m x in t co t t co t I Page

3 The energy iipate per cycle by the amping force can be repreente graphically a follow. Writing the velocity in the form x co t 1 in t x t The amping force become () cx c x Rearranging the above equation to x c 1 c x Area c c We recognize it a that of an ellipe with an x plotte along the vertical an horizontal axi, a hown above. It i of interet to examine the total (elatic & amping) reiting force that i meaure in an experiment. c x k x k x c x A plot of againt xi the ellipe. The energy iipate by amping i till the area encloe by the ellipe Page3

4 kx kx c x() t Loaing x Unloaing x k x, i zero. Damping propertie of material are lite in many ifferent way epening on the technical area to which they are applie. Of thee we lit two relative energy unit that have wie uage. irt of thee i pecific amping capacity, efine a the energy lo per cycle w ivie by the pick potential energy U. Specific amping capacity E Eo 1 Where Eo k Specific amping capacity c c 1 k k The econ quantity i the lo coefficient or pecific amping factor, i efine a the ratio of amping energy lo per raian w ivie by the peak potential or train energy U. w c U k or the cae of linear amping where the energy lo i proportional to the quare of train or amplitue, the hyterei curve i an ellipe. When the amping lo i not quaratic function of the train or amplitue, the hyterei curve i no longer an ellipe. Structural amping Page4

5 When material are cyclically tree, energy i iipate internally within the material itelf. Experiment by everal invetigator inicate that for mot tructural metal, uch a teel, aluminium, the energy iipate per cycle i inepenent of the frequency over a wie frequency range an proportional to the quare of the amplitue of vibration. Internal amping fitting thi claification i calle oli amping or tructural amping. Energy iipate by tructural amping may be written a w Where i contant with unit of force / iplacement. Uing the concept of equivalent vicou amping give or, C C eq eq Subtituting C eq for c, the ifferential equation of motion for a ytem with tructural amping may be written a m x x k x Complex tiffne: In the calculation of the flutter pee of air plane wing an tail urface, the concept of complex tiffne i ue. It i arrive at by auming the ocillation to be harmonic, which enable to be written a (t) i t mx k i x e Defining, the equation become k i t mx k 1 i x e factor. The quantity k ( 1 i i calle complex tiffne an i the tructural amping With the olution x e it, the teay tate amplitue become k m i k The amplitue of reonance i then k Comparing thi with the reonance amplitue of a ytem with vicou amping, k Page5

6 We conclue that with equal amplitue at reonance, the tructural amping factor i equal to twice the vicou amping factor Page6