ME 321 Kinematics and Dynamics of Machines S. Lambert Winter 2002

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1 ME 31 Kiemaic ad Dyamic o Machie S. Lamber Wier 6.. Forced Vibraio wih Dampig Coider ow he cae o orced vibraio wih dampig. Recall ha he goverig diereial equaio i: m && c& k F() ad ha we will aume ha he eciaio orce i a harmoic ucio: o ha: or, i ormalized orm: where F ) F co ( m&& c& k F co && ζ & k m c ζ m F m co or he udamped cae, we develop he eady-ae oluio uig he mehod o udeermied coeicie. We epec he oluio o be a harmoic ucio wih he ame requecy a he eciaio orce, ad ukow ampliude. Due o he preece o dampig, we alo have a phae lag,. The aumed oluio i hereore: p ( ) co( ) To impliy ubeque maipulaio, we ca rewrie hi a: where ( ) co i 1 a Subiuig hi io our diereial equaio give: p ad ( i b co) & ( ) p && ( ) p ( co i ) 96

2 ME 31 Kiemaic ad Dyamic o Machie S. Lamber Wier 97 Subiue hi io he goverig diereial equaio o ge: ( ) ( ) i co ζ ζ Sice hi mu be rue or all, we ca e he coeicie o co ad i equal o zero: ( ) ( ) ζ ζ Solvig or ad, we ge: ( ) ( ) ( ) ( ) ( ) ζ ζ ζ We ca ow cover back o a ampliude,, ad phae agle,, or he paricular oluio, where: ( ) ( ) 1 a ζ ζ Noe ha hee parameer are he ame a hoe or he udamped cae whe he dampig i e o zero. Noice alo ha hey do o deped o he iiial codiio, ice hey repree he eady-ae oluio. The oal oluio or a uderdamped yem he become: ( ) ( ) φ ζ e d co i ) ( To obai he iegraio coa ad φ i erm o he iiial codiio ad v, we ubiue hee iiial codiio io hi equaio ad i derivaive. The value o he iegraio coa will deped o he eady-ae oluio, ad ge a lile mey. They are: ( ) ( ) ζ φ φ i co co a i co 1 v d

3 ME 31 Kiemaic ad Dyamic o Machie S. Lamber Wier Foruaely, we are uually iereed primarily i he eady-ae repoe: ad. However, we mu ir aiy ourelve ha he raie repoe ha bee damped ou. Eample 6.4: Deermie he eady-ae repoe (ampliude ad phae agle) or a ma-prig damper yem ha ha he ollowig properie: F 1 N, m 1 kg, ζ.1, 1-1, ad 5-1. Wha i he oal repoe or a iiial diplaceme o.5 m ad o iiial velociy? The reul or he eady-ae oluio are give i Figure 6.1. The combied reul are give i Figure p Figure 6.1: Seady-ae repoe or eample

4 ME 31 Kiemaic ad Dyamic o Machie S. Lamber Wier c.1.5 p Figure 6.13: Toal repoe or eample 6.4. I i commo o ormalize he eady-ae reul a ollow. The ampliude,, i ormalized by he diplaceme caued by he applied orce, F, acig direcly o he prig ie, k: 5 F k k m Deiig he requecy raio, r /, hi give: or ( ) ( ζ ) 1 ζ ( 1 r ) ( r) Similarly, he phae lag,, ca be wrie i erm o r: 99

5 ME 31 Kiemaic ad Dyamic o Machie S. Lamber Wier a 1 ζr 1 r Thee reul are ploed i Figure Figure 6.14: Normalized eady-ae vibraio repoe. 1

6 ME 31 Kiemaic ad Dyamic o Machie S. Lamber Wier Peak Repoe Noe ha he peak repoe (maimum ) doe o acually occur a he aural requecy. We ca deermie he requecy raio, r, a which hi occur by akig he derivaive o he ormalized ampliude wih repec o r ad eig i o zero. I we do hi, we id ha he requecy raio a which he ampliude o he repoe i a maimum i equal o: or r ma r ma 1 ζ Noe ha hi epreio i oly valid or ζ 1. For value o he dampig raio above hi rage, here i o maimum oher ha a r. The value o he peak repoe i: k F 1 1 ζ ζ 11