EN40: Dyais ad Vibratios Hoework 6: Fored Vibratios Due Friday April 6, 018 Shool of Egieerig Brow Uiversity 1. The vibratio isolatio syste show i the figure has 0kg, k 19.8 kn / 1.59 kns / If the base vibrates haroially with a aplitude of 1 ad frequey of 100Hz, what is the steady-state aplitude of vibratio of the platfor (i.e. the ass )? k y(t) x(t) We just eed to fid the right forulas to use ad substitute ubers. We have that k / 31.46 rad / s ζ / ( k) 1 The agifiatio is 1 + ( ζ / ) M 0.1 (1 / ) + ( ζ / ) The vibratio aplitude is therefore X0 KMY0 MY0 0.1 [ POINTS]
. Both systes i the figure are subjeted to a fore with aplitude 1 kn ad frequey equal to the udaped atural frequey of the sprig-ass syste ( ). The vibratio aplitude of syste B is easured to be 1. What is the vibratio aplitude of syste A? F(t)F 0 si( t) F(t)F 0 si( t) k k Α Β The forula for aplitude is X0 KM ( /, ζ) F0. Whe the agifiatio M 1/ ( ζ ) where ζ / ( k). Reovig a dashpot does ot hage the udaped atural frequey ad halves ζ. The aplitude will double. 3. I this (hard!) proble we will aalyze the behavior of the ati-resoat vibratio isolatio syste itrodued i Hoework 5. The syste is illustrated i the figure. Assue that the base vibrates vertially with a displaeet yt ( ) Y0 sit. Our goal is to alulate a forula for the steady-state vertial otio xt () of the platfor, ad to opare the behavior of this syste with the stadard base exited sprig-ass-daper desig for a isolatio syste. x(t) k,l 1 1 y(t) L/ L/ L 1 L θ 3.1 Draw free body diagras showig the fores atig o the ass 1 ad the pedulu assebly (see the figure). 1 F S F D T T R y R x
3. Usig geoetry, fid a expressio for the aeleratio of ass i ters of θ, y ad their tie derivatives (as well as relevat geoetri ostats) (e.g. by writig dow a forula for the positio vetor relative to a fixed origi ad differetiatig it). Show that if θ ad its tie derivatives are sall the result a be approxiated by d y d θ a + L j Show also that d x d y d L 1 θ The positio vetor of is r Los θi+ ( y+ L/ + Lsi θ) j Differetiate twie with respet to tie to get d θ os d θ L si d y L d θ si d θ a θ + θ i+ + θ + osθ j Usig the approxiatios osθ 1 siθ θ ad egletig squared or higher order produts of θ ad its tie derivatives we get the result stated. Siilarly x y L L 1 siθ Differetiate this twie d x d y dθ d θ L 1 siθ + osθ Use the sae approxiatio to get the stated result. 3.3 For the pedulu, write dow F a ad M 0 about the eter of ass, i ters of reatio fores show i your FBD. Use the approxiatio i 3. for the aeleratio. For we have d y d θ Rxi+ ( Ry + T) j+ + L j RL x siθ Ry Los θ TL ( 1+ L)osθ 0 [ POINTS]
3.4 Write dow F a for ass 1, ad hee use 3.3 ad the seod of 3. to show that (if if θ ad its tie derivatives are sall) the L d x dx L L d y dy 1+ + + kx 1+ + + ky L L 1 1 L1 Fa gives d x d 1 T k( x y L) ( x y) We a solve (3.3) for T to get L d y d θ T + L L 1 Fro. L d y L d x d y T + L 1 L1 Hee substitute for T ad rearrage to get L d x dx L L d y dy 1+ + + kx kl + 1+ + + ky L L 1 1 L1 3.5 Show that the equatio a be re-arraged ito the for 1 d x dx d y dy x y + ζ λ ζ + + + ad show that ζ kl 1 L L1+ L λ L 1 1 L + L 1 1+ L k( L 1 1+ L)/ L1 ( ) ( ) We a re-write the equatio as
L 1 1+ L d x dx L 1 1+ L L ( L1+ L) d y dy + + x + + y k kl 1 kl 1 ( L 1 1 L ) + k Hee ζ kl 1 L L1+ L λ L 1 1 L + L 1 1+ L k( L 1 1+ L)/ L1 ( ) ( ) 3.6 Suppose that the base is subjeted to haroi exitatio y Y 0 sit. Show (usig alulus ad the double-agle forula osψ sit+ siψ ost si( t+ ψ) ) that d y dy + + y Y 0 1 sit+ ost λ ζ λ ζ λ ζ 1 ζ / Y0 1 si( t ) ta + + y y 1 λ ( / ) Hee use the Case IV solutio to differetial equatios to show that the steady state solutio for x has the for xt ( ) X0si( t+ φ) X0 M( /, ζ, λ) Y0 ad give a forula for the agifiatio fator M. Substitutig for y ad evaluatig the derivatives shows λ d y ζ dy λ ζ + + y Y 0 1 sit+ ost We a re-write this as λ 1 ζ λ ζ Y0 1 si t + + ost λ ζ λ ζ 1 + 1 + Defiig
λ 1 ζ osψ λ ζ λ ζ 1 + 1 + siψ We a use the double agle forula osψ sit siψ ost si( t ψ) stated. We a regard the equatio + + to get the aswer d y dy + + y Y 0 1 + si( t+ y) λ ζ λ ζ As a ase IV EOM with λ ζ K 1 + Y 0 F 0 The solutio follows fro the forula sheet, ad the agifiatio is M λ ζ 1 + ζ 1 + 3.7. Plot a graph of M as a futio of 0 < / < 6 for λ 0,for values of ζ 0.0,0.05,0.1,0. (o the sae plot). This graph shows the agifiatio for the stadard vibratio isolatio syste, sie λ 0 orrespods to a pedulu with zero ass it should look the sae as the Case V agifiatio graph disussed i lass. For opariso, plot a seod graph of M as a futio of / for λ 0.6 (a ati-resoat isolator),for values of ζ 0.0,0.05,0.1,0.
Magifiatio for stadard vibratio isolator Magifiatio for atiresoat vibratio isolator M 10 1 10 0 0.0 0.05 0.1 0. M 10 1 10 0 0.0 0.05 0.1 0. 10-1 10-1 0 1 3 4 5 / 0 1 3 4 5 / 3.8 What is the frequey orrespodig to the ati-resoae (the iiu value of M), i ters of λ, (give a approxiate solutio for ζ << 1 )? What is (approxiately) the sallest vibratio aplitude (i ters of λζ)?, λ The iiu will our whe 1 0 / λ. The orrespodig vibratio aplitude is ζ λ M 1 ζ 1 + λ λ If λ is ot lose to 1, the M ζ 1 λ λ [ POINTS] 3.9 For what rage of frequey (i ters of λ, ) does the pedulu syste give better perforae tha the sipler sprig-ass-daper syste? The agifiatio for the ati-resoat isolator is equal to that of the ovetioal syste whe This gives λ ζ ζ 1 + 1+
λ 1 1 λ λ The ati-resoat syste is better tha the ovetioal syste for below this value. It oly isolates vibatios if / >, however. 3.10 What sort of appliatio would be best suited for a ati-resoat vibratio isolator? [ POINTS] The isolator is oly useful if the exitatio is lose to a haroi otio at a fixed frequey - if the base will ove with a rage of frequeies, the isolator will blok the oes lose to the atiresoae, but ot the others. Atiresoat isolators are ofte used betwee the rotor blade assebly of a heliopter ad the heliopter body, for exaple (beause the exitatio frequey is set by the rotor agular speed) [ POINTS]
4 A ubalaed wid-turbie is idealized as a rotor-exited sprigass syste as show i the figure. The ass represets the tower, ad 0 represets the obied ass of the three rotor blades. The sprig ad daper represet the stiffess ad eergy dissipatio i the tower. The rotor is ubalaed beause its eter of ass is a sall distae Y 0 away fro the axle. The total ass ( + ) of the syste is 5000kg. 0 k t Y 0 0 The figure shows the results of a free vibratio experiet o the turbie. 4.1 Use the data provided to deterie the followig quatities: (a) The vibratio period The period is 0.5 se (b) The log dereet The log dereet is 1 log 0.5 1.6 0.0 () The udaped atural frequey 4π + d The udaped atural frequey is 1.8 rad / s T
(d) The dapig fator The dapig fator is ζ δ 4π + δ 0. (e) The sprig stiffess k k 4100 kn / (f) The dashpot oeffiiet ζ ζ k k 18000 Ns / 8 6 4 Displaeet () 0 - -4-6 -8 0 5 10 15 0 5 30 35 40 45 50 Tie (s) 4. The figure shows the easured displaeet of the syste durig operatio. The blades have a radius of 40, ad the total ass of the syste ( + 0 ) is 5000kg. Assuig that the rotor a be balaed by addig ass to the tip of oe blade, estiate the ass that ust be added to balae the rotor. We kow that the vibratio aplitude of the ubalaed rotor is X 0 0 Y0 0 / + (1 / ) + ( ζ / ) We a use the vibratio easureet to estiate the produt Y 0 0:
Fro the graph, we see that the period is about 6 se so 1 rad/s, so usig the ubers fro 4.1 0 1/1.8 0.0061 X0 Y0 Y 0 0 5000 (1 1/1.8 ) + ( 0. /1.8) 5000 Sie the easured aplitude is about 6, we olude that 0Y0 5000kg. We wat to ove the COM bak to the eter of the rotor reall that the COM is (1 / M) r ii so the required ass at the blade tip is 5000/40 65 kg. [4 POINTS]