x(t) y(t) c c F(t) F(t) EN40: Dynamics and Vibrations Homework 6: Forced Vibrations Due Friday April 5, 2018

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1 EN40: Dynais and Vibrations Hoewor 6: Fored Vibrations Due Friday April 5, 2018 Shool of Engineering Brown University 1. The vibration isolation syste shown in the figure has =20g, = 19.8 N / = Ns / The base vibrates haronially with an aplitude of 1 and angular frequeny ω. 1.1 What is the value of ω that will ause the platfor (the ass ) to vibrate with the greatest aplitude? What is the orresponding vibration aplitude? 1.2 What is the lowest value of ω for whih the vibration isolator is effetive (i.e. the aplitude of the platfor is less than the aplitude of the base)? y(t) x(t) 2. Systes A and B in the figure shown are subjeted to the sae haroni fore. The steady state aplitude of vibration of syste A is easured to be 1. What is the aplitude of vibration of syste B? A B 2

2 3. This IEEE transations on bioedial engineering paper desribes a vibration experient designed to easure the ass and viso-elasti properties of a ell. The ell is plaed on a piezoeletri vibrating platfor, and an atoi fore irosope is used to easure the displaeent of the top of the ell. The authors idealize the ell as a spring-ass-daper syste, and extrat values for the stiffness, dashpot oeffiient, and ass for live and dead ells fro their experiental data. Using a stati test, they easure a stiffness of 0.1 N/ for a live ell, and 0.2N/ for a dead one. Their data for the aplitude ratio (the vibration aplitude of the AFM tip divided by the vibration aplitude of the substrate) and phase lag (related to the tie lag θ between the zero rossing of the substrate and the zero rossing of the AFM tip and the period T as φ = 2 πθ /T ) are shown in the tables below. Live Cells Frequeny Aplitude Phase Lead (rad) (Hz) Ratio Dead Cells Frequeny Aplitude Phase lead (rad) (Hz) Ratio

3 3.1 The paper reports the following values for the ell ass and dashpot oeffiient fro their data. Live ell: = 10.5x10-12 g =0.393x10-6 Ns/ Dead ell: = 12.5x10-12 g =0.7x10-6 Ns/ The paper does not opare the preditions of the odel (equations 8 and 9 in the paper) with the experiental data, however, so we will try this. Calulate the values of daping oeffiient ζ, ω n and hene use the standard solutions for a baseexited spring-ass syste to plot (on the sae graph) the predited aplitude ratio (this is M in the engn40 notation) and the experiental data. Do a siilar seond plot for the phase. Don t forget to onvert frequeny fro Hz to rad/s. Don t worry if the theory and experient don t agree I ouldn t get it to wor either. There is no need to subit MATLAB ode for this proble, the graphs are suffiient. 3.2 We an attept to get better estiates for and. The estiates fro the paper for and suggest that the exitation frequeny is uh less than the natural frequeny. If this is the ase, we an derive siplified forulas for the aplitude ratio and phase that ae it easier to fit nubers to the data. Show that the forulas for M and φ for a base-exited spring-ass syste an be expressed in ters of,, and as ( ω / ) 1 ( ω/ )( ω / ) M = φ = tan /2 2 2 (1 ω / ) ( ω/ ) 1 ω / + ( ω/ ) + Hene, show that for ω << / the aplitude ratio an be approxiated by (use MATLAB to tae the Taylor series, or do it by hand) A 2 1 A + 0 ω while the phase an be approxiated by 3 φ ω Use 3.2 and the experiental data to estiate values for and for live and dead ell (Find a way to plot the data to get a straight line relationship between the y axis and the frequeny, so you an estiate fro the slope of the aplitude plot, and fro the slope of the phase plot. you will find the data does not give very good straight lines, so you will only be able to get approxiate values)

4 Displaeent () Frequeny (Hz) 4 As part of the airworthiness ertifiation proess, the rotating parts of a jet engine are prevented fro turning, and the engine is subjeted to an external horizontal haroni fore Ft ( ) = F0 sinωt with aplitude F 0 = 250N. The aplitude X 0 of the steady-state horizontal vibration xt ( ) = X0 sin( ωt+ φ) of the engine is easured. x(t) The easured displaeent aplitude X 0 is shown in the figure as a funtion of frequeny (in yles/se). 4.1 Assuing that the engine and its ounting are idealized as a spring-ass-daper syste (with light daping), use the graph provided to estiate values for the following quantities 0 Rotor (prevented fro rotating) (a) The natural frequeny of vibration of the engine (give both the frequeny in yles per seond and the angular frequeny) (b) The daping fator ζ. (Use the pea. Note that the graph shows the displaeent aplitude, not agnifiation M) () The spring stiffness (use the displaeent at very low frequeny) (d) The total ass + 0 (e) The dashpot oeffiient

5 4.2 During operation, the engine spins at 9550 rp. An aeleroeter ounted on the outside of the engine easures a haroni aeleration with aplitude 10 /s 2. What is the aplitude of the displaeent? Aeleroeter Y What is the engine speed (in rp) at whih the steady-state displaeent aplitude will be a axiu? ωt What is the steady-state displaeent aplitude when the engine runs at the speed in 4.3?