and ζ in 1.1)? 1.2 What is the value of the magnification factor M for system A, (with force frequency ω = ωn

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1 EN40: Dynais and Vibrations Hoework 6: Fored Vibrations, Rigid Body Kineatis Due Friday April 7, 017 Shool of Engineering Brown University 1. Syste A in the figure is ritially daped. The aplitude of the fore is 1 kn, and the frequeny of the fore is equal to the undaped natural frequeny of the spring-ass syste ( ω = ωn ). Its aplitude of vibration is easured to be What is the value of ζ for syste A? The syste is ritially daped, so ζ = 1 F(t)=F 0 sin( ω t) F(t)=F 0 sin( ω t) k k k Α Β 1. What is the value of the agnifiation fator M for syste A, (with fore frequeny ω = ωn and ζ in 1.1)? The forula for M fro the notes is 1 1 M = M = ζω ( 1 ω / ωn ) + ωn 1.3 Use 1. and the given fore aplitude and vibration aplitude to find a value for the spring stiffness in syste A. (you an use the forulas for vibration aplitude fro the notes, you don t have to re-derive the, but you are weloe to do so as a review if you like!) The forula for vibration aplitude is X0 = KMF0. We know that K=1/k so 1 1 F X0 = F0 k = = = 500 kn / k X

2 1.4 Syste B is exited with the sae fore (i.e. the fores in ating on A and B have the sae aplitude and frequeny). What is the vibration aplitude of B? For syste B, M k 1 1 ωn = ω / ωn = ζ = = k 1 1 = = = 5 ζω ( 1 ω / ω ) ( 1 1/ n + ) + ω n X = 0 MF = k = 5 [3 POINTS]. The goal of this proble is to design a vibration isolation syste (i.e. reoend values for the spring stiffness, ass, and the dashpot oeffiient in the idealization shown) with the following speifiations: (a) The syste should return to equilibriu as quikly as possible following a disturbane (b) The aplitude of vibration of the platfor should be less than 10% of the aplitude of the base for all frequenies exeeding 100Hz () The defletion of the platfor when an objet is plaed on it should be iniized (d) The ass of the syste annot exeed 0kg. k y(t) s(t).1 What value of daping fator ζ is needed to eet ondition (a)? We need ritial daping ζ = 1. We know that if the syste is disturbed, its otion will have the for x( t) = C+ {( x0 C) + [ v0 + ωn( x0 C) ] t} exp( ωnt) What does this tell us about hoosing - only whether we should ake ω n big or sall). ω n needs to be big ω n needed to eet ondition (a)? (It won t tell us the value of ω n

3 .3 Write down the forula for the agnifiation M of the base-exited syste with ζ fro.1, in ters of ω / ω n. Hene, use ondition (b) to show that 1/ { 1+ ( ω / ωn ) } < 0.1 1/ ( 1 ω / ωn) + ( ω / ωn) 00π for all frequenies ω > 00π. Dedue that ωn (you an use Mupad to solve the equation if you like), and hene use. to selet the value for ω n that will best eet the design speifiation. The seond ondition says that 1/ X { 1+ ( ςω / ωn ) } 0 = < 0.1 Y 1/ 0 ( 1 ω / ωn) + ( ςω / ωn) for all frequenies ω > 00π. We already know that ζ = 1 so this ondition an be solved for / n we find (see the upad below) that ω / ωn ω ω : Sine we want to axiize ω n to eet the first ondition, we have that ωn = (00 π) / = rad/s [3 POINTS].4 What does ondition () tell us about the hoosing the spring stiffness k? (It won t tell us the value, only whether k should be big or sall). The defletion of the platfor when an objet of ass M is plaed on it is Mg / k. To iniize this, we want to use the stiffest possible spring.

4 .5 Use the solution to.3,.4 and ondition (d) to selet the best values for k and, and use.1 to deterine the neessary value for. k Sine ω n =, if we axiize k we also have to axiize. So we pik =0kg, whih gives k = ωn = 0 (31.485) = 19.8 kn / ζ = / ( k) = ζ k = 0 = 1.59 kns / [3 POINTS] 3. In this publiation Rapaport and Beah show that it is possible to detet the diaeter of a iro-bead by easuring its fored vibration response in a highly loalized flutuating agneti field (this would provide a way to sort iron-sale partiles or ells, for exaple). The figure (fro their publiation) illustrates the operating priniple. A loalized agneti doain is reated on a substrate by e-bea lithography. A paraagneti partile with radius R and ass is plaed on the substrate. The doain attrats the partiles to its enter, and behaves like a spring with stiffness k. The position of the doain is varied haronially ydw = Y0 sin ωt by applying an external AC field to the substrate. The partiles are in a fluid, and so are subjeted to a visous drag fore FD = 6πµ Rv, where µ is the fluid visosity, and v is the speed of the partile. 3.1 The figure shows ore learly how the authors idealize their devie. The dashpot represents visous fluid fores ating on the bead: the dashpot oeffiient is related to the bead radius and fluid visosity by = 6πµ R. The otion of the end of the spring ydw ( t ) = Y0 sin ωt represents the flutuation in the agneti field. Use Newton s law to find the equation of otion for the syste, and show that the equation of otion for the sphere is a Case IV differential equation 1 d x ζ dx + + x = C + Ky t ωn dt ωn dt with ωn = k / ζ = K = 1 k DW () x(t) y DW (t) k,l 0 A free body diagra for the ass is shown F D F S

5 d x Newton s law gives = F S FD dt The spring fore law gives FS = k( ydw x) dx FD =, whih gives dt d x dx + + x= Y 0 sinωt k dt k dt For this proble ωn = k / ζ = K = 1 k [ POINTS] 3. Estiate values for ζ, ωn for the following values of paraeters: Bead radius 1 3 Fluid visosity (water) 10 Ns / Bead ass density 1800 kg/ 3 6 Stiffness k 4 10 N / Substituting nubers gives ζ = 54.3, ω n = 3000 rad / s 3.3 Use 3. and the solution for Case IV vibrations to plot the expeted aplitude of vibration of the bead as a funtion of frequeny ω for an exitation aplitudey 0 =1, for frequenies in the range 10 < ω < Note that (beause the syste is very heavily daped) the resonant peak is not visible. Beause the resonant peak is not visible, the usual trik of looking for resonane to learn soething about a syste does not work for this appliation. The authors needed to find soe other way to detet the bead size fro their easureents. The figure is shown (see Mupad file for details).

6 3.4 Use the Case 4 steady-state solutions to show that the displaeent x an be expressed as xt = A( ω)sin ωt B( ω)osωt and find forulas for A( ω), B( ω ) for the paraeters given in 3.. (you will need to use the forula for the phase shift φ fro the notes, and also use the double angle forula sin( θ + θ ) = sinθ osθ + osθ sinθ ). Plot A( ω), B( ω ) for a displaeent aplitude X 0 =1 and 10 < ω < The peak in B( ω ) depends on ζ, and so an be used to deterine the bead radius (although the authors refer to the peak as resonane in their paper they are not using the usual definition of resonane). You don t need to subit Mupad/Matlab ode for this proble, just inlude a opy of your graphs [ POINTS] The solution for ase 4 is xt = YX 0 0sin( ωt+ φ) 1 ζω / ω X0 = tanφ = n 1 ω / ω 1 ω / ω ( / ) n n + ζω ωn The double angle forula gives Note that ( sin ω os φ os ω sin φ ) x= YX 0 0 t + t

7 xt = YX 0 0sin( ωt+ φ) ( 1 ω / ωn ) ζω / ω sinφ = n osφ = 1 ω / ωn + ( ζω / ωn) 1 ω / ωn + ( ζω / ωn) Hene xt = YX 0 0sin( ωt+ φ) ( 1 ω / ωn ) Y0 ζω / ω B( ω) = n A( ω) = 1 ω / ωn + ( ζω / ωn) 1 ω / ωn + ( ζω / ωn) [4 POINTS]