Problem 1. Problem Engineering Dynamics Problem Set 9--Solution. Find the equation of motion for the system shown with respect to:

Transcription

1 2.003 Egieerig Dyamics Problem Set 9--Solutio Problem 1 Fid the equatio of motio for the system show with respect to: a) Zero sprig force positio. Draw the appropriate free body diagram. b) Static equilibrium positio. Draw the appropriate free body diagram. c) Cocept questio: Does g eter ito the expressio for the udamped atural frequecy? Aswer. The acceleratio of gravity will ot eter ito the expressio for the atural frequecy, because the term ivolvig g i the EOM is a costat ad ot a fuctio of the motio coordiate, x(t). Problem 2 A thi hoop of mass m ad radius R is hagig from a kife edge (there is o slip betwee the hoop ad the kife edge). The force F(t) is always horizotal. a) Draw a free body diagram ad derive the equatio of motio of the hoop. b) Fid the liearized equatio of motio usig a small agle approximatio for θ (defie θ such that the static equilibrium positio is θ=0). c) Fid the udamped atural frequecy of the system. Cocept questio: Does g eter ito the expressio for the udamped atural frequecy? Aswer: I this case the acceleratio of gravity appears i a term i the EOM which ivolves the agle of the motio,. The atural frequecy expressio will iclude g. 1

2 Problem 3 Try to come up with a geeral rule that will predict whe the acceleratio of gravity will appear i the expressio for the atural frequecy. Perhaps compare the systems of problems 1 ad 2 to illustrate your aswer. Problem 4 For the system i problem 1, let K=10,000 N/m, M=0.633 kg, x o =0.1m, v o =10 m/s ad the dampig ratio =0.05. a. Fid a expressio for x(t) i terms of the iitial coditios. Express x(t) i the form x( t ) Acos( t ). b. Sketch x(t) versus time. c. Compute the ratio of the damped to the udamped atural frequecy for dampig ratios of (i) 5%, (ii) 10% ad (iii) 20% of the critical dampig. Cocept questio: If i a experimet this system was give a iitial velocity ad observed to decay i amplitude of vibratio by 50% i 10 cycles of vibratio, ca you estimate the dampig ratio of the system. i , ii..01, iii,.02? Aswer: The simple rule of thumb says that if 50% is the umber of cycles of motio required to decay i amplitude by 50%, the the dampig ratio is give by % Problem 5 For the values of M, K used i the previous problem, compute the steady state amplitude ad phase agle of the respose to a harmoic force specified as F(t ) F cos( t ), where F o =10N ad the dampig ratio is 5% of critical. Do this computatio for three values of ω/ ω = (i) 0.5, (ii) 1.0 ad (iii) 3.0. i. Cocept questio: At which of the three excitatio frequecy ratios will the respose magitude be greatest. i. 0.05, ii. 1.0, iii 3.0? Aswer: The respose will be greatest at resoace, whe 1.0. o 2

3 Problem 6 A block of wood is suspeded by two strigs, as show i the Figure below. The strigs are separated by 10.0 cm. ad are 28 cm i legth. a= 14.0 cm, b=6.0 cm, ad c=1.8 cm. a) Determie the mass momet of iertia with respect to the vertical axis of rotatio which passes through the ceter of mass of the block. b) Fid a equatio of motio for small torsioal oscillatios about the vertical axis which passes through the ceter of mass. Use the direct method i which you sum the exteral momets to fid the equatio of motio. Liearize the equatio of motio for small oscillatios. c) Fid expressios for T ad V, the kietic ad potetial eergies of the block i terms of the rotatioal velocity ad the agle of rotatio of the block. Cocept questio: Ca all of the kietic eergy 1 2 be accouted for by a expressio of the formt I? 2 zz Aswer: No, there will be some vertical traslatioal velocity of the ceter of mass ad this body is ot rotatig. I zz will ot eter ito the solutio. Problem 7 Cosider the same system as i problem 6. a) Fid the kietic ad potetial eergy expressios for the block whe it swigs i the plae of the paper as show i the figure. This meas there is o torsioal motio about the vertical axis passig through the ceter of mass. b) Use the cocept of coservatio of total eergy to fid the equatio of motio. Liearize the EOM ad fid the atural frequecy. c) Assume that we coduct a experimet with this system. It is give a iitial agular deflectio of 0.2 radias ad released. Over time the oscillatio amplitude becomes smaller due to 3

4 dampig. After 5 cycles of vibratio the maximum agle of motio has reduced to 0.08 radias. What is the approximate dampig ratio of the system. d) Cocept questio: Will the expressio for T have a term of the formt 1 2 I? 2 zz Aswer: This body has traslatioal velocity oly. There is o rotatio ad therefore o kietic eergy of the form show i the questio above. 4

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